Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 864865 11 pageshttpdxdoiorg1011552013864865

Research ArticleA Fourth-Order Block-Grid Method forSolving Laplacersquos Equation on a Staircase Polygonwith Boundary Functions in 119862119896120582

A A Dosiyev and S Cival Buranay

Department of Mathematics Eastern Mediterranean University Gazimagusa North Cyprus Mersin 10 Turkey

Correspondence should be addressed to A A Dosiyev adiguzeldosiyevemuedutr

Received 30 April 2013 Accepted 11 May 2013

Academic Editor Allaberen Ashyralyev

Copyright copy 2013 A A Dosiyev and S Cival Buranay This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The integral representations of the solution around the vertices of the interior reentered angles (on the ldquosingularrdquo parts) areapproximated by the composite midpoint rule when the boundary functions are from 119862

4120582 0 lt 120582 lt 1 These approximations

are connected with the 9-point approximation of Laplacersquos equation on each rectangular grid on the ldquononsingularrdquo part of thepolygon by the fourth-order gluing operator It is proved that the uniform error is of order 119874(ℎ

4+ 120576) where 120576 gt 0 and ℎ is the

mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between the approximate and the exact solutions in each ldquosingularrdquo part 119874((ℎ

4+ 120576)119903

1120572119895minus119901

119895) order is obtained here 119903

119895is the distance from the current point to the vertex in question and 120572

119895120587

is the value of the interior angle of the 119895th vertex Numerical results are given in the last section to support the theoretical results

1 Introduction

In the last two decades among different approaches to solvethe elliptic boundary value problems with singularities aspecial emphasis has been placed on the construction ofcombined methods in which differential properties of thesolution in different parts of the domain are used (see [1 2]and references therein)

In [2ndash7] a new combined difference-analytical methodcalled the block-grid method (BGM) is proposed for thesolution of the Laplace equation on polygons when theboundary functions on the sides causing the singular verticesare given as algebraic polynomials of the arc length In theBGM the given polygon is covered by a finite number ofoverlapping sectors around the singular vertices (ldquosingularrdquoparts) and rectangles for the part of the polygon which lies ata positive distance from these vertices (ldquononsingularrdquo part)The special integral representation in each ldquosingularrdquo partis approximated and they are connected by the appropriateorder gluing operator with the finite difference equationsused in the ldquononsingularrdquo part of the polygon

In [8 9] the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices in the BGM was removed It was assumedthat the boundary function on each side of the polygon isgiven from the Holder classes 119862119896120582

0 lt 120582 lt 1 and on theldquononsingularrdquo part the 5-point scheme is used when 119896 = 2

[8] and the 9-point scheme is used when 119896 = 6 [9] For the5-point scheme a simple linear interpolation with 4 pointsis used For the 9-point scheme an interpolation with 31points is used to construct a gluing operator connectingthe subsystems Moreover to connect the quadrature nodeswhich are at a distance of less than 4ℎ from boundary of thepolygon a special representation of the harmonic functionthrough the integrals of Poisson type for a half plane is used(see [9])

In this paper the BGM is developed for the Dirichletproblem when the boundary function on each side of thepolygon is from 119862

4120582 by using the 9-point scheme on theldquononsingularrdquo part with 16-point gluing operator for allquadrature nodes including those near the boundary The

2 Abstract and Applied Analysis

paper is organized as follows in Section 2 the boundaryvalue problem and the integral representations of the exactsolution in each ldquosingularrdquo part are given In Section 3 tosupport the aim of the paper a Dirichlet problem on therectangle for the known exact solution from 119862

119896120582 119896 = 3 4 issolved using the 9-point scheme and the numerical results areillustrated In Section 4 the system of block-grid equationsand the convergence theorems are given In Section 5 a highlyaccurate approximation of the coefficient of the leadingsingular term of the exact solution (stress intensity factor) isgiven In Section 6 the method is illustrated for solving theproblem in L-shaped polygonwith the corner singularityTheconclusions are summarized in Section 7

2 Dirichlet Problem on a Staircase Polygon

Let 119866 be an open simply connected polygon 120574119895 119895 =

1 2 119873 its sides including the ends enumerated coun-terclockwise 120574 = 120574

1cup sdot sdot sdot cup 120574

119873the boundary of 119866 and 120572

119895120587

(120572119895= 12 1 32 2) the interior angle formed by the sides

120574119895minus1

and 120574119895 (120574

0= 120574

119873) Denote by 119860

119895= 120574

119895minus1cap 120574

119895the vertex of

the 119895th angle and by 119903119895 120579

119895a polar system of coordinates with a

pole in119860119895 where the angle 120579

119895is taken counterclockwise from

the side 120574119895

We consider the boundary value problem

Δ119906 = 0 on 119866 119906 = 120593119895(119904) on 120574

119895 1 le 119895 le 119873 (1)

whereΔ equiv 1205972120597119909

2+120597

2120597119910

2 120593

119895is a given function on 120574

119895of the

arc length 119904 taken along 120574 and 120593119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 that

is 120593119895has the fourth-order derivative on 120574

119895 which satisfies a

Holder condition with exponent 120582At some vertices119860

119895 (119904 = 119904

119895) for 120572

119895= 12 the conjugation

conditions

120593(2119902)

119895minus1(119904

119895) = (minus1)

119902120593(2119902)

119895(119904

119895) 119902 = 0 1 (2)

are fulfilled For the remaining vertices 119860119895 the values of 120593

119895minus1

and 120593119895at 119860

119895might be different Let 119864 be the set of all 119895 (1 le

119895 le 119873) for which 120572119895

= 12 or 120572119895= 12 but (2) is not fulfilled

In the neighborhood of 119860119895 119895 isin 119864 we construct two fixed

block sectors 119879119894

119895= 119879

119895(119903

119895119894) sub 119866 119894 = 1 2 where 0 lt 119903

1198952lt 119903

1198951lt

min119904119895+1

minus119904119895 119904

119895minus119904

119895minus1 119879

119895(119903) = (119903

119895 120579

119895) 0 lt 119903

119895lt 119903 0 lt 120579

119895lt

120572119895120587Let (see [10])

1205931198950(119905) = 120593

119895(119904

119895+ 119905) minus 120593

119895(119904

119895)

1205931198951(119905) = 120593

119895minus1(119904

119895minus 119905) minus 120593

119895minus1(119904

119895)

(3)

119876119895(119903

119895 120579

119895) = 120593

119895(119904

119895) +

(120593119895minus1

(119904119895) minus 120593

119895(119904

119895)) 120579

119895

120572119895120587

+1

120587

1

sum

119896=0

int

120590119895119896

0

119910119895120593119895119896

(119905120572119895) 119889119905

(119905 minus (minus1)119896119909119895)2

+ 1199102

119895

(4)

where

119909119895= 119903

1120572119895

119895cos(

120579119895

120572119895

) 119910119895= 119903

1120572119895

119895sin(

120579119895

120572119895

)

120590119895119896

=10038161003816100381610038161003816119904119895+1minus119896

minus 119904119895minus119896

10038161003816100381610038161003816

1120572119895

(5)

The function119876119895(119903

119895 120579

119895) is harmonic on119879

1

119895and satisfies the

boundary conditions in (1) on 120574119895minus1

cap 1198791

119895and 120574

119895cap 119879

1

119895 119895 isin 119864

except for the point 119860119895when 120593

119895minus1(119904

119895) = 120593

119895(119904

119895)

We formally set the value of 119876119895(119903

119895 120579

119895) and the solution

119906 of the problem (1) at the vertex 119860119895equal to (120593

119895minus1(119904

119895) +

120593119895(119904

119895))2 119895 isin 119864Let

119877119895(119903 120579 120578)

=1

120572119895

1

sum

119896=0

(minus1)119896119877((

119903

1199031198952

)

1120572119895

120579

120572119895

(minus1)119896 120578

120572119895

)

119895 isin 119864

(6)

where

119877 (119903 120579 120578) =1 minus 119903

2

2120587 (1 minus 2119903 cos (120579 minus 120578) + 1199032)(7)

is the kernel of the Poisson integral for a unit circle

Lemma 1 (see [10]) The solution 119906 of the boundary valueproblem (1) can be represented on 119879

2

119895 119881

119895 119895 isin 119864 in the form

119906 (119903119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ int

120572119895120587

0

119877119895(119903

119895 120579

119895 120578) (119906 (119903

1198952 120578) minus 119876

119895(119903

1198952 120578)) 119889120578

(8)

where 119881119895is the curvilinear part of the boundary of 1198792

119895 and

119876119895(119903

119895 120579

119895) is the function defined by (4)

3 9-Point Solution on Rectangles

Let Π = (119909 119910) 0 lt 119909 lt 119886 0 lt 119910 lt 119887 be a rectanglewith 119886119887 being rational 120574

119895 119895 = 1 2 3 4 the sides including

the ends enumerated counterclockwise starting from the leftside (120574

0equiv 120574

4 120574

5equiv 120574

1) and 120574 = cup

4

119895=1120574119895the boundary of Π

We consider the boundary value problem

Δ119906 = 0 on Π

119906 = 120593119895

on 120574119895 119895 = 1 2 3 4

(9)

where 120593119895is the given function on 120574

119895

Definition 2 One says that the solution 119906 of the problem (9)belongs to 119862

4120582(Π) if

120593119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 119895 = 1 2 3 4 (10)

Abstract and Applied Analysis 3

and at the vertices 119860119895= 120574

119895minus1cap 120574

119895the conjugation conditions

120593(2119902)

119895= (minus1)

119902120593(2119902)

119895minus1 119902 = 0 1 (11)

are satisfied

Remark 3 From Theorem 31 in [11] it follows that the classof functions 1198624120582

(Π) is wider than 1198624120582

(Π)

Let ℎ gt 0 with 119886ℎ ge 2 119887ℎ ge 2 integers We assignto Π

ℎ a square net on Π with step ℎ obtained with the lines119910 = 0 ℎ 2ℎ Let 120574ℎ

119895be a set of nodes on the interior of 120574

119895

and let

120574ℎ

119895= 120574

119895cap 120574

119895+1 120574

ℎ= cup

4

119895=1(120574

ℎ

119895cup 120574

ℎ

119895)

Πℎ

= Πℎcup 120574

ℎ

(12)

We consider the system of finite difference equations

119906ℎ= 119861119906

ℎon Π

ℎ

119906ℎ= 120593

119895on 120574

ℎ

119895 119895 = 1 2 3 4

(13)

where

119861119906 (119909 119910)

equiv(119906 (119909 + ℎ 119910) + 119906 (119909 119910 + ℎ) + 119906 (119909 minus ℎ 119910) + 119906 (119909 119910 minus ℎ))

5

+ ( ((119906 (119909 + ℎ 119910 + ℎ) + 119906 (119909 minus ℎ 119910 + ℎ)

+ 119906 (119909 minus ℎ 119910 minus ℎ) + 119906 (119909 + ℎ 119910 minus ℎ)))

times 20minus1)

(14)

On the basis of the maximum principle the uniquesolvability of the system of finite difference equations (13)follows (see [12 Chapter 4])

Everywhere below we will denote constants which areindependent of ℎ and of the cofactors on their right by119888 119888

0 119888

1 generally using the same notation for different

constants for simplicity

Theorem 4 Let 119906 be the solution of problem (9) If 119906 isin

1198624120582

(Π) then

maxΠℎ

1003816100381610038161003816119906ℎ minus 1199061003816100381610038161003816 le ch4 (15)

where 119906ℎis the solution of the system (13)

Proof For the proof of this theorem see [13]

LetΠ1015840= (119909 119910) minus025 lt 119909 lt 025 0 lt 119910 lt 1 and let 1205741015840

be the boundary of Π1015840 We consider the Dirichlet problem

Δ119906 = 0 on Π1015840

119906 = V on 1205741015840

(16)

0 20 40 60 80 100 120 14011

12

13

14

15

16

17

119862400003

11986240003

1198624003

1198623055

1198623066

1198623075

119862308

Rϱ

hminus1 = 2ϱ

Figure 1 Dependence of the approximate solutions for the bound-ary functions from 119862

119896120582

where V = 119903119896+120582 cos(119896 + 120582)120579 119903 = radic1199092 + 1199102 0 lt 120582 lt 1 is the

exact solution of this problem which is from 119862119896120582

(Π1015840

)We solve the problem (16) by approximating 9-point

scheme when 119896 = 3 4 for the different values of 120582In Figure 1 the order of numerical convergence

R984858

Πℎ =

maxΠℎ10038161003816100381610038161199062minus984858 minus 119906

1003816100381610038161003816

maxΠℎ10038161003816100381610038161199062minus(984858+1) minus 119906

1003816100381610038161003816

(17)

of the 9-point solution 119906ℎ for different ℎ = 2

minus984858 and 984858 =

4 5 6 7 is demonstratedThese results show that the order ofnumerical convergence when the exact solution 119906 isin 119862

119896120582(Π)

depends on 119896 and 120582 and is119874(ℎ4)when 119896 = 4 which supports

estimation (15) Moreover this dependence also requires theuse of fourth-order gluing operator for all quadrature nodesin the construction of the system of block-grid equationswhen the given boundary functions are from the Holderclasses 1198624120582

4 System of Block-Grid Equations

In addition to the sectors 1198791

119895and 119879

2

119895(see Section 2) in the

neighborhood of each vertex 119860119895 119895 isin 119864 of the polygon 119866 we

construct two more sectors 1198793

119895and 119879

4

119895 where 0 lt 119903

1198954lt 119903

1198953lt

1199031198952 119903

1198953= (119903

1198952+ 119903

1198954)2 and 119879

3

119896cap 119879

3

119897= 0 119896 = 119897 119896 119897 isin 119864 and let

119866119879= 119866 (cup

119895isin1198641198794

119895)

We cover the given solution domain (a staircase polygon)by the finite number of sectors 119879

1

119895 119895 isin 119864 and rectangles

Π119896

sub 119866119879 119896 = 1 2 119872 as is shown in Figure 2 for

the case of 119871-shaped polygon where 119895 = 1119872 = 4 (seealso [2]) It is assumed that for the sides 119886

1119896and 119886

2119896of

Π119896the quotient 119886

1119896119886

2119896is rational and 119866 = (cup

119872

119896=1Π

119896) cup

(cup119895isin119864

1198793

119895) Let 120578

119896be the boundary of the rectangleΠ

119896 let119881

119895be

4 Abstract and Applied Analysis

minus1 minus05 0 05 1minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Π1

Π212057810

12057820

119886

119887119888

119889119890

120579

1198601 1205741

1205740

11987941

11987931

11987921

11987911

Π3

Π4

12057820

12057830

12057830

12057840

119903

11990311

11990312

11990313 11990314

119866119878

Figure 2 Description of BGM for the L-shaped domain

the curvilinear part of the boundary of the sector 1198792

119895 and let

119905119896119895

= 120578119896cap 119879

3

119895 We choose a natural number 119899 and define

the quantities 119899(119895) = max4 [120572119895119899] 120573

119895= 120572

119895120587119899(119895) and

120579119898

119895= (119898 minus 12)120573

119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881

119895

we take the points (1199031198952 120579

119898

119895) 1 le 119898 le 119899(119895) and denote the

set of these points by 119881119899

119895 We introduce the parameter ℎ isin

(0 12060004] where 120600

0is a gluing depth of the rectanglesΠ

119896 119896 =

1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872

withmaximal possible step ℎ119896le minℎmin119886

1119896 119886

21198966 such

that the boundary 120578119896lies entirely on the grid lines Let Πℎ

119896be

the set of grid nodes on Π119896 let 120578ℎ

119896be the set of nodes on 120578

119896

and let Πℎ

119896= Π

ℎ

119896cup 120578

ℎ

119896 We denote the set of nodes on the

closure of 120578119896cap 119866

119879by 120578

ℎ

1198960 the set of nodes on 119905

119896119895by 119905

ℎ

119896119895 and

the set of remaining nodes on 120578119896by 120578

ℎ

1198961

Let

120596ℎ119899

= (cup119872

119896=1120578ℎ

1198960) cup (cup

119895isin119864119881

119899

119895)

119866ℎ119899

119879= 120596

ℎ119899cup (cup

119872

119896=1Π

ℎ

119896)

(18)

Let 120593 = 120593119895119873

119895=1 where 120593

119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 is

the given function in (1) We use the matching operator 1198784

at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906

ℎ 120593) at the point 119875 isin 120596

ℎ119899 is expressed linearly in termsof the values of 119906

ℎat the points 119875

119896of the grid constructed on

Π119896(119875)

(119875 isin Π119896(119875)

) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)

119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906

ℎ 0) has the representation

1198784(119906

ℎ 0) = sum

0le119897le15

120585119897119906ℎ119897 (19)

where 119906ℎ119896

= 119906ℎ(119875

119896)

120585119897ge 0 sum

0le119897le15

120585119897= 1 (20)

119906 minus 1198784(119906 120593) = 119874 (ℎ

4) (21)

Let 120596ℎ119899

119868sub 120596

ℎ119899 be the set of such points 119875 isin 120596ℎ119899

for which all points 119875119897in expression (19) are in cup

119872

119896=1Π

ℎ

119896 If

some points 119875119897in (19) emerge through the side 120574

119898 then the

set of such points 119875 is denoted by 120596ℎ119899

119863 According to the

construction of 1198784 in [14] the expression 1198784(119906

ℎ 120593) at each

point 119875 isin 120596ℎ119899

= 120596ℎ119899

119868cup 120596

ℎ119899

119863can be expressed as follows

1198784(119906

ℎ 120593)

=

1198784119906ℎ 119875 isin 120596

ℎ119899

119868

1198784(119906

ℎminus

3

sum

119896=0

119886119896Re 119911119896) + (

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(22)

where

119886119896=

1

119896

119889119896120593119898(119904)

119889119904119896

100381610038161003816100381610038161003816100381610038161003816119904=119904119875

119896 = 0 1 2 3 (23)

and 119904119875corresponds to such a point119876 isin 120574

119898for which the line

119875119876 is perpendicular to 120574119898

Let

119876119895= 119876

119895(119903

119895 120579

119895) 119876

119902

1198952= 119876

119895(119903

1198952 120579

119902

119895) (24)

The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876

120576

119895and 119876

119902120576

1198952of the

quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876

120576

119895minus 119876

119895

10038161003816100381610038161003816le 119888

1120576

10038161003816100381610038161003816119876

119902120576

1198952minus 119876

119902

1198952

10038161003816100381610038161003816le 119888

2120576 (25)

where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888

2are constants

independent of 120576Consider the system of linear algebraic equations

119906120576

ℎ= 119861119906

120576

ℎon Π

ℎ

119896

119906120576

ℎ= 120593

119898on 120578

ℎ

1198961cap 120574

119898

119906120576

ℎ(119903

119895 120579

119895)

= 119876120576

119895+ 120573

119895

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

times 119877119895(119903

119895 120579

119895 120579

119902

119895) on (119903

119895 120579

119895) isin 119905

ℎ

119896119895

119906120576

ℎ= 119878

4119906120576

ℎon 120596

ℎ119899

(26)

where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864

Abstract and Applied Analysis 5

Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2

minus984858 and 120576 = 5 times 10minus13

(2minus984858 119899) 120577

120576

ℎ119866119873119878 R

984858

119866119873119878

(2minus4 60) 1609 times 10

minus815577

(2minus5 170) 1033 times 10

minus9

(2minus5 130) 1191 times 10

minus916690

(2minus6 150) 7136 times 10

minus11

(2minus5 140) 1136 times 10

minus916259

(2minus6 170) 6991 times 10

minus11

(2minus6 100) 2169 times 10

minus1017096

(2minus7 130) 1269 times 10

minus11

Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =

2minus984858 and 120576 = 5 times 10

minus13

(2minus984858 119899) 120577

120576

ℎ119866119878 R

984858

119866119878

(2minus4 100) 1931 times 10

minus816078

(2minus5 150) 1182 times 10

minus9

(2minus5 130) 1294 times 10

minus916789

(2minus6 150) 7708 times 10

minus11

(2minus5 140) 1312 times 10

minus917967

(2minus6 170) 7304 times 10

minus11

(2minus6 100) 2389 times 10

minus1018164

(2minus7 130) 1315 times 10

minus11

Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) 120577

120576

ℎ119866119873119878 120577

120576

ℎ119866119878 Iteration

(16 70) 1139 times 10minus8

1572 times 10minus8

22

(32 170) 1033 times 10minus9

1184 times 10minus9

23

(64 170) 6990 times 10minus11

7304 times 10minus11

24

(128 200) 8628 times 10minus12

8833 times 10minus12

25

Let 119879lowast

119895= 119879

119895(119903

lowast

119895) be the sector where 119903lowast

119895= (119903

1198952+119903

1198953)2 119895 isin

119864 and let 119906120576ℎ(119903

1198952 120579

119902

119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution

values of the system (26) on 119881ℎ

119895(at the quadrature nodes)

The function

119880120576

ℎ(119903

119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ 120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) (119906

120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

(27)

defined on 119879lowast

119895 is called an approximate solution of the

problem (1) on the closed block 1198793

119895 119895 isin 119864

Definition 5 The system (26) and (27) is called the system ofblock-grid equations

Theorem 6 There is a natural number 1198990 such that for all

119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution

Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899

0and 120590 such that for all 119899 ge 119899

0

max(119903119895120579119895)isin119879

3

119895

120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) le 120590 lt 1 (28)

The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)

Theorem 7 There exists a natural number 1198990 such that for all

119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)

where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid

max119866ℎ119899

119879

1003816100381610038161003816119906120576

ℎminus 119906

1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816le 119888

119901(ℎ

4+ 120576) on 119879

3

119895

(31)

for integer 1120572119895when 119901 ge 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895

(32)

for any 1120572119895 if 0 le 119901 lt 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895 119860

119895

(33)

for noninteger 1120572119895 when 119901 gt 1120572

119895 Everywhere 0 le 119902 le 119901 119906

is the exact solution of the problem (1) and119880120576

ℎ(119903

119895 120579

119895) is defined

by formula (27)

Proof Let

120585120576

ℎ= 119906

120576

ℎminus 119906 (34)

where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866

ℎ119899

119879

of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576

ℎsatisfies the system of difference equations

120585120576

ℎ= 119861120585

120576

ℎ+ 119903

1

ℎon Π

ℎ

119896

120585120576

ℎ= 0 on 120578

ℎ

1198961

120585120576

ℎ(119903

119895 120579

119895) = 120573

119895

119899(119895)

sum

119902=1

120585120576

ℎ(119903

1198952 120579

119902

119895) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

+ 1199032

119895ℎ (119903

119895 120579

119895) isin 119905

ℎ

119896119895

120585120576

ℎ= 119878

4120585120576

ℎ+ 119903

3

ℎon 120596

ℎ119899

(35)

6 Abstract and Applied Analysis

Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 3895 times 10minus7

3895 times 10minus7

(32 170) 4627 times 10minus8

4627 times 10minus8

(64 170) 1124 times 10minus9

3125 times 10minus9

(128 200) 2214 times 10minus10

2233 times 10minus10

Table 5 In119866119878 the minimum errors of the derivatives over the pairs

(ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 9663 times 10minus6

9663 times 10minus6

(32 170) 9653 times 10minus6

9653 times 10minus6

(64 170) 9649 times 10minus6

9649 times 10minus6

(128 200) 9648 times 10minus6

9648 times 10minus6

where 1 le 119896 le 119872 119895 isin 119864

1199031

ℎ= 119861119906 minus 119906 on cup

119872

119896=1Π

ℎ

119896 (36)

1199032

119895ℎ= 120573

119895

119899(119895)

sum

119902=1

(119906 (1199031198952 120579

119902

119895) minus 119876

119902120576

1198952) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

minus (119906 minus 119876120576

119895) on cup

119872

119896=1(cup

119895isin119864119905ℎ

119896119895)

(37)

1199033

ℎ

=

1198784119906 minus 119906 on 120596

ℎ119899

119868

1198784(119906 minus

3

sum

119896=0

119886119896Re 119911119896) minus (119906 minus

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(38)

On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows

The function 119880120576

ℎ(119903

119895 120579

119895) given by formula (27) defined

on the closed sector 119879lowast

119895 119895 isin 119864 where 119903

lowast

119895= (119903

1198952+ 119903

1198953)2

and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879

2

119895 119881

119895 119895 isin 119864 and then the

difference function 120577120576

ℎ(119903

119895 120579

119895) = 119880

120576

ℎ(119903

119895 120579

119895)minus119906(119903

119895 120579

119895) is defined

on 119879lowast

119895 119895 isin 119864 and

120577120576

ℎ(119903

lowast

119895 0) = 120577

120576

ℎ(119903

lowast

119895 120572

119895120587) = 0 119895 isin 119864 (39)

On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge

max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain

10038161003816100381610038161003816120577120576

ℎ(119903

119895 120579

119895)10038161003816100381610038161003816le 119888 (ℎ

4+ 120576) on 119879

lowast

119895 119895 isin 119864 (40)

Furthermore the function 120577120576

ℎ(119903

119895 120579

119895) continuous on 119879

lowast

119895is a

solution of the following Dirichlet problem

Δ120577120576

ℎ= 0 on 119879

lowast

119895

120577120576

ℎ= 0 on 120574

119898cap 119879

lowast

119895 119898 = 119895 minus 1 119895

120577120576

ℎ(119903

lowast

119895 120579

119895) = 119880

120576

ℎ(119903

lowast

119895 120579

119895) minus 119906 (119903

lowast

119895 120579

119895) 0 le 120579

119895le 120572

119895120587

(41)

Since 1198793

119895sub 119879

lowast

119895 on the basis of (39) and (40) from Lemma

612 in [16] inequalities (31)ndash(33) of Theorem 7 follow

5 Stress Intensity Factor

Let in the condition 120593119895isin 119862

4120582(120574

119895) the exponent 120582 be such

that

120572119895(4 + 120582) = 0 2120572

119895(4 + 120582) = 0 (42)

where sdot is the symbol of fractional partThese conditions forthe given 120572

119895can be fulfilled by decreasing 120582

On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879

lowast

119895 119895 isin 119864 as follows

119906 (119909119895 119910

119895) = (119909

119895 119910

119895) +

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911

+

119899120572119895

sum

119896=1

120591(119895)

119896119903119896120572119895

119895sin

119896120579119895

120572119895

(43)

where 119899120572119895

= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909

119895+119894119910

119895 120583(119895)

119896

and 120591(119895)

119896are some numbers and (119909

119895 119910

119895) isin 119862

4120582(119879

2

119895) is the

harmonic on 1198792

119895 By taking 120579

119895= 120572

1198951205872 from the formula

(43) it follows that the coefficient 120591(119895)1

which is called the stressintensity factor can be represented as

120591(119895)

1= lim

119903119895rarr0

1

1199031120572119895

119895

(119906 (119909119895 119910

119895) minus (119909

119895 119910

119895)

minus

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911)

(44)

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

paper is organized as follows in Section 2 the boundaryvalue problem and the integral representations of the exactsolution in each ldquosingularrdquo part are given In Section 3 tosupport the aim of the paper a Dirichlet problem on therectangle for the known exact solution from 119862

119896120582 119896 = 3 4 issolved using the 9-point scheme and the numerical results areillustrated In Section 4 the system of block-grid equationsand the convergence theorems are given In Section 5 a highlyaccurate approximation of the coefficient of the leadingsingular term of the exact solution (stress intensity factor) isgiven In Section 6 the method is illustrated for solving theproblem in L-shaped polygonwith the corner singularityTheconclusions are summarized in Section 7

2 Dirichlet Problem on a Staircase Polygon

Let 119866 be an open simply connected polygon 120574119895 119895 =

1 2 119873 its sides including the ends enumerated coun-terclockwise 120574 = 120574

1cup sdot sdot sdot cup 120574

119873the boundary of 119866 and 120572

119895120587

(120572119895= 12 1 32 2) the interior angle formed by the sides

120574119895minus1

and 120574119895 (120574

0= 120574

119873) Denote by 119860

119895= 120574

119895minus1cap 120574

119895the vertex of

the 119895th angle and by 119903119895 120579

119895a polar system of coordinates with a

pole in119860119895 where the angle 120579

119895is taken counterclockwise from

the side 120574119895

We consider the boundary value problem

Δ119906 = 0 on 119866 119906 = 120593119895(119904) on 120574

119895 1 le 119895 le 119873 (1)

whereΔ equiv 1205972120597119909

2+120597

2120597119910

2 120593

119895is a given function on 120574

119895of the

arc length 119904 taken along 120574 and 120593119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 that

is 120593119895has the fourth-order derivative on 120574

119895 which satisfies a

Holder condition with exponent 120582At some vertices119860

119895 (119904 = 119904

119895) for 120572

119895= 12 the conjugation

conditions

120593(2119902)

119895minus1(119904

119895) = (minus1)

119902120593(2119902)

119895(119904

119895) 119902 = 0 1 (2)

are fulfilled For the remaining vertices 119860119895 the values of 120593

119895minus1

and 120593119895at 119860

119895might be different Let 119864 be the set of all 119895 (1 le

119895 le 119873) for which 120572119895

= 12 or 120572119895= 12 but (2) is not fulfilled

In the neighborhood of 119860119895 119895 isin 119864 we construct two fixed

block sectors 119879119894

119895= 119879

119895(119903

119895119894) sub 119866 119894 = 1 2 where 0 lt 119903

1198952lt 119903

1198951lt

min119904119895+1

minus119904119895 119904

119895minus119904

119895minus1 119879

119895(119903) = (119903

119895 120579

119895) 0 lt 119903

119895lt 119903 0 lt 120579

119895lt

120572119895120587Let (see [10])

1205931198950(119905) = 120593

119895(119904

119895+ 119905) minus 120593

119895(119904

119895)

1205931198951(119905) = 120593

119895minus1(119904

119895minus 119905) minus 120593

119895minus1(119904

119895)

(3)

119876119895(119903

119895 120579

119895) = 120593

119895(119904

119895) +

(120593119895minus1

(119904119895) minus 120593

119895(119904

119895)) 120579

119895

120572119895120587

+1

120587

1

sum

119896=0

int

120590119895119896

0

119910119895120593119895119896

(119905120572119895) 119889119905

(119905 minus (minus1)119896119909119895)2

+ 1199102

119895

(4)

where

119909119895= 119903

1120572119895

119895cos(

120579119895

120572119895

) 119910119895= 119903

1120572119895

119895sin(

120579119895

120572119895

)

120590119895119896

=10038161003816100381610038161003816119904119895+1minus119896

minus 119904119895minus119896

10038161003816100381610038161003816

1120572119895

(5)

The function119876119895(119903

119895 120579

119895) is harmonic on119879

1

119895and satisfies the

boundary conditions in (1) on 120574119895minus1

cap 1198791

119895and 120574

119895cap 119879

1

119895 119895 isin 119864

except for the point 119860119895when 120593

119895minus1(119904

119895) = 120593

119895(119904

119895)

We formally set the value of 119876119895(119903

119895 120579

119895) and the solution

119906 of the problem (1) at the vertex 119860119895equal to (120593

119895minus1(119904

119895) +

120593119895(119904

119895))2 119895 isin 119864Let

119877119895(119903 120579 120578)

=1

120572119895

1

sum

119896=0

(minus1)119896119877((

119903

1199031198952

)

1120572119895

120579

120572119895

(minus1)119896 120578

120572119895

)

119895 isin 119864

(6)

where

119877 (119903 120579 120578) =1 minus 119903

2

2120587 (1 minus 2119903 cos (120579 minus 120578) + 1199032)(7)

is the kernel of the Poisson integral for a unit circle

Lemma 1 (see [10]) The solution 119906 of the boundary valueproblem (1) can be represented on 119879

2

119895 119881

119895 119895 isin 119864 in the form

119906 (119903119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ int

120572119895120587

0

119877119895(119903

119895 120579

119895 120578) (119906 (119903

1198952 120578) minus 119876

119895(119903

1198952 120578)) 119889120578

(8)

where 119881119895is the curvilinear part of the boundary of 1198792

119895 and

119876119895(119903

119895 120579

119895) is the function defined by (4)

3 9-Point Solution on Rectangles

Let Π = (119909 119910) 0 lt 119909 lt 119886 0 lt 119910 lt 119887 be a rectanglewith 119886119887 being rational 120574

119895 119895 = 1 2 3 4 the sides including

the ends enumerated counterclockwise starting from the leftside (120574

0equiv 120574

4 120574

5equiv 120574

1) and 120574 = cup

4

119895=1120574119895the boundary of Π

We consider the boundary value problem

Δ119906 = 0 on Π

119906 = 120593119895

on 120574119895 119895 = 1 2 3 4

(9)

where 120593119895is the given function on 120574

119895

Definition 2 One says that the solution 119906 of the problem (9)belongs to 119862

4120582(Π) if

120593119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 119895 = 1 2 3 4 (10)

Abstract and Applied Analysis 3

and at the vertices 119860119895= 120574

119895minus1cap 120574

119895the conjugation conditions

120593(2119902)

119895= (minus1)

119902120593(2119902)

119895minus1 119902 = 0 1 (11)

are satisfied

Remark 3 From Theorem 31 in [11] it follows that the classof functions 1198624120582

(Π) is wider than 1198624120582

(Π)

Let ℎ gt 0 with 119886ℎ ge 2 119887ℎ ge 2 integers We assignto Π

ℎ a square net on Π with step ℎ obtained with the lines119910 = 0 ℎ 2ℎ Let 120574ℎ

119895be a set of nodes on the interior of 120574

119895

and let

120574ℎ

119895= 120574

119895cap 120574

119895+1 120574

ℎ= cup

4

119895=1(120574

ℎ

119895cup 120574

ℎ

119895)

Πℎ

= Πℎcup 120574

ℎ

(12)

We consider the system of finite difference equations

119906ℎ= 119861119906

ℎon Π

ℎ

119906ℎ= 120593

119895on 120574

ℎ

119895 119895 = 1 2 3 4

(13)

where

119861119906 (119909 119910)

equiv(119906 (119909 + ℎ 119910) + 119906 (119909 119910 + ℎ) + 119906 (119909 minus ℎ 119910) + 119906 (119909 119910 minus ℎ))

5

+ ( ((119906 (119909 + ℎ 119910 + ℎ) + 119906 (119909 minus ℎ 119910 + ℎ)

+ 119906 (119909 minus ℎ 119910 minus ℎ) + 119906 (119909 + ℎ 119910 minus ℎ)))

times 20minus1)

(14)

On the basis of the maximum principle the uniquesolvability of the system of finite difference equations (13)follows (see [12 Chapter 4])

Everywhere below we will denote constants which areindependent of ℎ and of the cofactors on their right by119888 119888

0 119888

1 generally using the same notation for different

constants for simplicity

Theorem 4 Let 119906 be the solution of problem (9) If 119906 isin

1198624120582

(Π) then

maxΠℎ

1003816100381610038161003816119906ℎ minus 1199061003816100381610038161003816 le ch4 (15)

where 119906ℎis the solution of the system (13)

Proof For the proof of this theorem see [13]

LetΠ1015840= (119909 119910) minus025 lt 119909 lt 025 0 lt 119910 lt 1 and let 1205741015840

be the boundary of Π1015840 We consider the Dirichlet problem

Δ119906 = 0 on Π1015840

119906 = V on 1205741015840

(16)

0 20 40 60 80 100 120 14011

12

13

14

15

16

17

119862400003

11986240003

1198624003

1198623055

1198623066

1198623075

119862308

Rϱ

hminus1 = 2ϱ

Figure 1 Dependence of the approximate solutions for the bound-ary functions from 119862

119896120582

where V = 119903119896+120582 cos(119896 + 120582)120579 119903 = radic1199092 + 1199102 0 lt 120582 lt 1 is the

exact solution of this problem which is from 119862119896120582

(Π1015840

)We solve the problem (16) by approximating 9-point

scheme when 119896 = 3 4 for the different values of 120582In Figure 1 the order of numerical convergence

R984858

Πℎ =

maxΠℎ10038161003816100381610038161199062minus984858 minus 119906

1003816100381610038161003816

maxΠℎ10038161003816100381610038161199062minus(984858+1) minus 119906

1003816100381610038161003816

(17)

of the 9-point solution 119906ℎ for different ℎ = 2

minus984858 and 984858 =

4 5 6 7 is demonstratedThese results show that the order ofnumerical convergence when the exact solution 119906 isin 119862

119896120582(Π)

depends on 119896 and 120582 and is119874(ℎ4)when 119896 = 4 which supports

estimation (15) Moreover this dependence also requires theuse of fourth-order gluing operator for all quadrature nodesin the construction of the system of block-grid equationswhen the given boundary functions are from the Holderclasses 1198624120582

4 System of Block-Grid Equations

In addition to the sectors 1198791

119895and 119879

2

119895(see Section 2) in the

neighborhood of each vertex 119860119895 119895 isin 119864 of the polygon 119866 we

construct two more sectors 1198793

119895and 119879

4

119895 where 0 lt 119903

1198954lt 119903

1198953lt

1199031198952 119903

1198953= (119903

1198952+ 119903

1198954)2 and 119879

3

119896cap 119879

3

119897= 0 119896 = 119897 119896 119897 isin 119864 and let

119866119879= 119866 (cup

119895isin1198641198794

119895)

We cover the given solution domain (a staircase polygon)by the finite number of sectors 119879

1

119895 119895 isin 119864 and rectangles

Π119896

sub 119866119879 119896 = 1 2 119872 as is shown in Figure 2 for

the case of 119871-shaped polygon where 119895 = 1119872 = 4 (seealso [2]) It is assumed that for the sides 119886

1119896and 119886

2119896of

Π119896the quotient 119886

1119896119886

2119896is rational and 119866 = (cup

119872

119896=1Π

119896) cup

(cup119895isin119864

1198793

119895) Let 120578

119896be the boundary of the rectangleΠ

119896 let119881

119895be

4 Abstract and Applied Analysis

minus1 minus05 0 05 1minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Π1

Π212057810

12057820

119886

119887119888

119889119890

120579

1198601 1205741

1205740

11987941

11987931

11987921

11987911

Π3

Π4

12057820

12057830

12057830

12057840

119903

11990311

11990312

11990313 11990314

119866119878

Figure 2 Description of BGM for the L-shaped domain

the curvilinear part of the boundary of the sector 1198792

119895 and let

119905119896119895

= 120578119896cap 119879

3

119895 We choose a natural number 119899 and define

the quantities 119899(119895) = max4 [120572119895119899] 120573

119895= 120572

119895120587119899(119895) and

120579119898

119895= (119898 minus 12)120573

119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881

119895

we take the points (1199031198952 120579

119898

119895) 1 le 119898 le 119899(119895) and denote the

set of these points by 119881119899

119895 We introduce the parameter ℎ isin

(0 12060004] where 120600

0is a gluing depth of the rectanglesΠ

119896 119896 =

1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872

withmaximal possible step ℎ119896le minℎmin119886

1119896 119886

21198966 such

that the boundary 120578119896lies entirely on the grid lines Let Πℎ

119896be

the set of grid nodes on Π119896 let 120578ℎ

119896be the set of nodes on 120578

119896

and let Πℎ

119896= Π

ℎ

119896cup 120578

ℎ

119896 We denote the set of nodes on the

closure of 120578119896cap 119866

119879by 120578

ℎ

1198960 the set of nodes on 119905

119896119895by 119905

ℎ

119896119895 and

the set of remaining nodes on 120578119896by 120578

ℎ

1198961

Let

120596ℎ119899

= (cup119872

119896=1120578ℎ

1198960) cup (cup

119895isin119864119881

119899

119895)

119866ℎ119899

119879= 120596

ℎ119899cup (cup

119872

119896=1Π

ℎ

119896)

(18)

Let 120593 = 120593119895119873

119895=1 where 120593

119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 is

the given function in (1) We use the matching operator 1198784

at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906

ℎ 120593) at the point 119875 isin 120596

ℎ119899 is expressed linearly in termsof the values of 119906

ℎat the points 119875

119896of the grid constructed on

Π119896(119875)

(119875 isin Π119896(119875)

) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)

119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906

ℎ 0) has the representation

1198784(119906

ℎ 0) = sum

0le119897le15

120585119897119906ℎ119897 (19)

where 119906ℎ119896

= 119906ℎ(119875

119896)

120585119897ge 0 sum

0le119897le15

120585119897= 1 (20)

119906 minus 1198784(119906 120593) = 119874 (ℎ

4) (21)

Let 120596ℎ119899

119868sub 120596

ℎ119899 be the set of such points 119875 isin 120596ℎ119899

for which all points 119875119897in expression (19) are in cup

119872

119896=1Π

ℎ

119896 If

some points 119875119897in (19) emerge through the side 120574

119898 then the

set of such points 119875 is denoted by 120596ℎ119899

119863 According to the

construction of 1198784 in [14] the expression 1198784(119906

ℎ 120593) at each

point 119875 isin 120596ℎ119899

= 120596ℎ119899

119868cup 120596

ℎ119899

119863can be expressed as follows

1198784(119906

ℎ 120593)

=

1198784119906ℎ 119875 isin 120596

ℎ119899

119868

1198784(119906

ℎminus

3

sum

119896=0

119886119896Re 119911119896) + (

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(22)

where

119886119896=

1

119896

119889119896120593119898(119904)

119889119904119896

100381610038161003816100381610038161003816100381610038161003816119904=119904119875

119896 = 0 1 2 3 (23)

and 119904119875corresponds to such a point119876 isin 120574

119898for which the line

119875119876 is perpendicular to 120574119898

Let

119876119895= 119876

119895(119903

119895 120579

119895) 119876

119902

1198952= 119876

119895(119903

1198952 120579

119902

119895) (24)

The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876

120576

119895and 119876

119902120576

1198952of the

quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876

120576

119895minus 119876

119895

10038161003816100381610038161003816le 119888

1120576

10038161003816100381610038161003816119876

119902120576

1198952minus 119876

119902

1198952

10038161003816100381610038161003816le 119888

2120576 (25)

where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888

2are constants

independent of 120576Consider the system of linear algebraic equations

119906120576

ℎ= 119861119906

120576

ℎon Π

ℎ

119896

119906120576

ℎ= 120593

119898on 120578

ℎ

1198961cap 120574

119898

119906120576

ℎ(119903

119895 120579

119895)

= 119876120576

119895+ 120573

119895

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

times 119877119895(119903

119895 120579

119895 120579

119902

119895) on (119903

119895 120579

119895) isin 119905

ℎ

119896119895

119906120576

ℎ= 119878

4119906120576

ℎon 120596

ℎ119899

(26)

where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864

Abstract and Applied Analysis 5

Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2

minus984858 and 120576 = 5 times 10minus13

(2minus984858 119899) 120577

120576

ℎ119866119873119878 R

984858

119866119873119878

(2minus4 60) 1609 times 10

minus815577

(2minus5 170) 1033 times 10

minus9

(2minus5 130) 1191 times 10

minus916690

(2minus6 150) 7136 times 10

minus11

(2minus5 140) 1136 times 10

minus916259

(2minus6 170) 6991 times 10

minus11

(2minus6 100) 2169 times 10

minus1017096

(2minus7 130) 1269 times 10

minus11

Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =

2minus984858 and 120576 = 5 times 10

minus13

(2minus984858 119899) 120577

120576

ℎ119866119878 R

984858

119866119878

(2minus4 100) 1931 times 10

minus816078

(2minus5 150) 1182 times 10

minus9

(2minus5 130) 1294 times 10

minus916789

(2minus6 150) 7708 times 10

minus11

(2minus5 140) 1312 times 10

minus917967

(2minus6 170) 7304 times 10

minus11

(2minus6 100) 2389 times 10

minus1018164

(2minus7 130) 1315 times 10

minus11

Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) 120577

120576

ℎ119866119873119878 120577

120576

ℎ119866119878 Iteration

(16 70) 1139 times 10minus8

1572 times 10minus8

22

(32 170) 1033 times 10minus9

1184 times 10minus9

23

(64 170) 6990 times 10minus11

7304 times 10minus11

24

(128 200) 8628 times 10minus12

8833 times 10minus12

25

Let 119879lowast

119895= 119879

119895(119903

lowast

119895) be the sector where 119903lowast

119895= (119903

1198952+119903

1198953)2 119895 isin

119864 and let 119906120576ℎ(119903

1198952 120579

119902

119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution

values of the system (26) on 119881ℎ

119895(at the quadrature nodes)

The function

119880120576

ℎ(119903

119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ 120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) (119906

120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

(27)

defined on 119879lowast

119895 is called an approximate solution of the

problem (1) on the closed block 1198793

119895 119895 isin 119864

Definition 5 The system (26) and (27) is called the system ofblock-grid equations

Theorem 6 There is a natural number 1198990 such that for all

119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution

Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899

0and 120590 such that for all 119899 ge 119899

0

max(119903119895120579119895)isin119879

3

119895

120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) le 120590 lt 1 (28)

The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)

Theorem 7 There exists a natural number 1198990 such that for all

119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)

where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid

max119866ℎ119899

119879

1003816100381610038161003816119906120576

ℎminus 119906

1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816le 119888

119901(ℎ

4+ 120576) on 119879

3

119895

(31)

for integer 1120572119895when 119901 ge 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895

(32)

for any 1120572119895 if 0 le 119901 lt 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895 119860

119895

(33)

for noninteger 1120572119895 when 119901 gt 1120572

119895 Everywhere 0 le 119902 le 119901 119906

is the exact solution of the problem (1) and119880120576

ℎ(119903

119895 120579

119895) is defined

by formula (27)

Proof Let

120585120576

ℎ= 119906

120576

ℎminus 119906 (34)

where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866

ℎ119899

119879

of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576

ℎsatisfies the system of difference equations

120585120576

ℎ= 119861120585

120576

ℎ+ 119903

1

ℎon Π

ℎ

119896

120585120576

ℎ= 0 on 120578

ℎ

1198961

120585120576

ℎ(119903

119895 120579

119895) = 120573

119895

119899(119895)

sum

119902=1

120585120576

ℎ(119903

1198952 120579

119902

119895) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

+ 1199032

119895ℎ (119903

119895 120579

119895) isin 119905

ℎ

119896119895

120585120576

ℎ= 119878

4120585120576

ℎ+ 119903

3

ℎon 120596

ℎ119899

(35)

6 Abstract and Applied Analysis

Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 3895 times 10minus7

3895 times 10minus7

(32 170) 4627 times 10minus8

4627 times 10minus8

(64 170) 1124 times 10minus9

3125 times 10minus9

(128 200) 2214 times 10minus10

2233 times 10minus10

Table 5 In119866119878 the minimum errors of the derivatives over the pairs

(ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 9663 times 10minus6

9663 times 10minus6

(32 170) 9653 times 10minus6

9653 times 10minus6

(64 170) 9649 times 10minus6

9649 times 10minus6

(128 200) 9648 times 10minus6

9648 times 10minus6

where 1 le 119896 le 119872 119895 isin 119864

1199031

ℎ= 119861119906 minus 119906 on cup

119872

119896=1Π

ℎ

119896 (36)

1199032

119895ℎ= 120573

119895

119899(119895)

sum

119902=1

(119906 (1199031198952 120579

119902

119895) minus 119876

119902120576

1198952) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

minus (119906 minus 119876120576

119895) on cup

119872

119896=1(cup

119895isin119864119905ℎ

119896119895)

(37)

1199033

ℎ

=

1198784119906 minus 119906 on 120596

ℎ119899

119868

1198784(119906 minus

3

sum

119896=0

119886119896Re 119911119896) minus (119906 minus

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(38)

On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows

The function 119880120576

ℎ(119903

119895 120579

119895) given by formula (27) defined

on the closed sector 119879lowast

119895 119895 isin 119864 where 119903

lowast

119895= (119903

1198952+ 119903

1198953)2

and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879

2

119895 119881

119895 119895 isin 119864 and then the

difference function 120577120576

ℎ(119903

119895 120579

119895) = 119880

120576

ℎ(119903

119895 120579

119895)minus119906(119903

119895 120579

119895) is defined

on 119879lowast

119895 119895 isin 119864 and

120577120576

ℎ(119903

lowast

119895 0) = 120577

120576

ℎ(119903

lowast

119895 120572

119895120587) = 0 119895 isin 119864 (39)

On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge

max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain

10038161003816100381610038161003816120577120576

ℎ(119903

119895 120579

119895)10038161003816100381610038161003816le 119888 (ℎ

4+ 120576) on 119879

lowast

119895 119895 isin 119864 (40)

Furthermore the function 120577120576

ℎ(119903

119895 120579

119895) continuous on 119879

lowast

119895is a

solution of the following Dirichlet problem

Δ120577120576

ℎ= 0 on 119879

lowast

119895

120577120576

ℎ= 0 on 120574

119898cap 119879

lowast

119895 119898 = 119895 minus 1 119895

120577120576

ℎ(119903

lowast

119895 120579

119895) = 119880

120576

ℎ(119903

lowast

119895 120579

119895) minus 119906 (119903

lowast

119895 120579

119895) 0 le 120579

119895le 120572

119895120587

(41)

Since 1198793

119895sub 119879

lowast

119895 on the basis of (39) and (40) from Lemma

612 in [16] inequalities (31)ndash(33) of Theorem 7 follow

5 Stress Intensity Factor

Let in the condition 120593119895isin 119862

4120582(120574

119895) the exponent 120582 be such

that

120572119895(4 + 120582) = 0 2120572

119895(4 + 120582) = 0 (42)

where sdot is the symbol of fractional partThese conditions forthe given 120572

119895can be fulfilled by decreasing 120582

On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879

lowast

119895 119895 isin 119864 as follows

119906 (119909119895 119910

119895) = (119909

119895 119910

119895) +

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911

+

119899120572119895

sum

119896=1

120591(119895)

119896119903119896120572119895

119895sin

119896120579119895

120572119895

(43)

where 119899120572119895

= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909

119895+119894119910

119895 120583(119895)

119896

and 120591(119895)

119896are some numbers and (119909

119895 119910

119895) isin 119862

4120582(119879

2

119895) is the

harmonic on 1198792

119895 By taking 120579

119895= 120572

1198951205872 from the formula

(43) it follows that the coefficient 120591(119895)1

which is called the stressintensity factor can be represented as

120591(119895)

1= lim

119903119895rarr0

1

1199031120572119895

119895

(119906 (119909119895 119910

119895) minus (119909

119895 119910

119895)

minus

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911)

(44)

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

and at the vertices 119860119895= 120574

119895minus1cap 120574

119895the conjugation conditions

120593(2119902)

119895= (minus1)

119902120593(2119902)

119895minus1 119902 = 0 1 (11)

are satisfied

Remark 3 From Theorem 31 in [11] it follows that the classof functions 1198624120582

(Π) is wider than 1198624120582

(Π)

Let ℎ gt 0 with 119886ℎ ge 2 119887ℎ ge 2 integers We assignto Π

ℎ a square net on Π with step ℎ obtained with the lines119910 = 0 ℎ 2ℎ Let 120574ℎ

119895be a set of nodes on the interior of 120574

119895

and let

120574ℎ

119895= 120574

119895cap 120574

119895+1 120574

ℎ= cup

4

119895=1(120574

ℎ

119895cup 120574

ℎ

119895)

Πℎ

= Πℎcup 120574

ℎ

(12)

We consider the system of finite difference equations

119906ℎ= 119861119906

ℎon Π

ℎ

119906ℎ= 120593

119895on 120574

ℎ

119895 119895 = 1 2 3 4

(13)

where

119861119906 (119909 119910)

equiv(119906 (119909 + ℎ 119910) + 119906 (119909 119910 + ℎ) + 119906 (119909 minus ℎ 119910) + 119906 (119909 119910 minus ℎ))

5

+ ( ((119906 (119909 + ℎ 119910 + ℎ) + 119906 (119909 minus ℎ 119910 + ℎ)

+ 119906 (119909 minus ℎ 119910 minus ℎ) + 119906 (119909 + ℎ 119910 minus ℎ)))

times 20minus1)

(14)

On the basis of the maximum principle the uniquesolvability of the system of finite difference equations (13)follows (see [12 Chapter 4])

Everywhere below we will denote constants which areindependent of ℎ and of the cofactors on their right by119888 119888

0 119888

1 generally using the same notation for different

constants for simplicity

Theorem 4 Let 119906 be the solution of problem (9) If 119906 isin

1198624120582

(Π) then

maxΠℎ

1003816100381610038161003816119906ℎ minus 1199061003816100381610038161003816 le ch4 (15)

where 119906ℎis the solution of the system (13)

Proof For the proof of this theorem see [13]

LetΠ1015840= (119909 119910) minus025 lt 119909 lt 025 0 lt 119910 lt 1 and let 1205741015840

be the boundary of Π1015840 We consider the Dirichlet problem

Δ119906 = 0 on Π1015840

119906 = V on 1205741015840

(16)

0 20 40 60 80 100 120 14011

12

13

14

15

16

17

119862400003

11986240003

1198624003

1198623055

1198623066

1198623075

119862308

Rϱ

hminus1 = 2ϱ

Figure 1 Dependence of the approximate solutions for the bound-ary functions from 119862

119896120582

where V = 119903119896+120582 cos(119896 + 120582)120579 119903 = radic1199092 + 1199102 0 lt 120582 lt 1 is the

exact solution of this problem which is from 119862119896120582

(Π1015840

)We solve the problem (16) by approximating 9-point

scheme when 119896 = 3 4 for the different values of 120582In Figure 1 the order of numerical convergence

R984858

Πℎ =

maxΠℎ10038161003816100381610038161199062minus984858 minus 119906

1003816100381610038161003816

maxΠℎ10038161003816100381610038161199062minus(984858+1) minus 119906

1003816100381610038161003816

(17)

of the 9-point solution 119906ℎ for different ℎ = 2

minus984858 and 984858 =

4 5 6 7 is demonstratedThese results show that the order ofnumerical convergence when the exact solution 119906 isin 119862

119896120582(Π)

depends on 119896 and 120582 and is119874(ℎ4)when 119896 = 4 which supports

estimation (15) Moreover this dependence also requires theuse of fourth-order gluing operator for all quadrature nodesin the construction of the system of block-grid equationswhen the given boundary functions are from the Holderclasses 1198624120582

4 System of Block-Grid Equations

In addition to the sectors 1198791

119895and 119879

2

119895(see Section 2) in the

neighborhood of each vertex 119860119895 119895 isin 119864 of the polygon 119866 we

construct two more sectors 1198793

119895and 119879

4

119895 where 0 lt 119903

1198954lt 119903

1198953lt

1199031198952 119903

1198953= (119903

1198952+ 119903

1198954)2 and 119879

3

119896cap 119879

3

119897= 0 119896 = 119897 119896 119897 isin 119864 and let

119866119879= 119866 (cup

119895isin1198641198794

119895)

We cover the given solution domain (a staircase polygon)by the finite number of sectors 119879

1

119895 119895 isin 119864 and rectangles

Π119896

sub 119866119879 119896 = 1 2 119872 as is shown in Figure 2 for

the case of 119871-shaped polygon where 119895 = 1119872 = 4 (seealso [2]) It is assumed that for the sides 119886

1119896and 119886

2119896of

Π119896the quotient 119886

1119896119886

2119896is rational and 119866 = (cup

119872

119896=1Π

119896) cup

(cup119895isin119864

1198793

119895) Let 120578

119896be the boundary of the rectangleΠ

119896 let119881

119895be

4 Abstract and Applied Analysis

minus1 minus05 0 05 1minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Π1

Π212057810

12057820

119886

119887119888

119889119890

120579

1198601 1205741

1205740

11987941

11987931

11987921

11987911

Π3

Π4

12057820

12057830

12057830

12057840

119903

11990311

11990312

11990313 11990314

119866119878

Figure 2 Description of BGM for the L-shaped domain

the curvilinear part of the boundary of the sector 1198792

119895 and let

119905119896119895

= 120578119896cap 119879

3

119895 We choose a natural number 119899 and define

the quantities 119899(119895) = max4 [120572119895119899] 120573

119895= 120572

119895120587119899(119895) and

120579119898

119895= (119898 minus 12)120573

119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881

119895

we take the points (1199031198952 120579

119898

119895) 1 le 119898 le 119899(119895) and denote the

set of these points by 119881119899

119895 We introduce the parameter ℎ isin

(0 12060004] where 120600

0is a gluing depth of the rectanglesΠ

119896 119896 =

1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872

withmaximal possible step ℎ119896le minℎmin119886

1119896 119886

21198966 such

that the boundary 120578119896lies entirely on the grid lines Let Πℎ

119896be

the set of grid nodes on Π119896 let 120578ℎ

119896be the set of nodes on 120578

119896

and let Πℎ

119896= Π

ℎ

119896cup 120578

ℎ

119896 We denote the set of nodes on the

closure of 120578119896cap 119866

119879by 120578

ℎ

1198960 the set of nodes on 119905

119896119895by 119905

ℎ

119896119895 and

the set of remaining nodes on 120578119896by 120578

ℎ

1198961

Let

120596ℎ119899

= (cup119872

119896=1120578ℎ

1198960) cup (cup

119895isin119864119881

119899

119895)

119866ℎ119899

119879= 120596

ℎ119899cup (cup

119872

119896=1Π

ℎ

119896)

(18)

Let 120593 = 120593119895119873

119895=1 where 120593

119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 is

the given function in (1) We use the matching operator 1198784

at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906

ℎ 120593) at the point 119875 isin 120596

ℎ119899 is expressed linearly in termsof the values of 119906

ℎat the points 119875

119896of the grid constructed on

Π119896(119875)

(119875 isin Π119896(119875)

) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)

119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906

ℎ 0) has the representation

1198784(119906

ℎ 0) = sum

0le119897le15

120585119897119906ℎ119897 (19)

where 119906ℎ119896

= 119906ℎ(119875

119896)

120585119897ge 0 sum

0le119897le15

120585119897= 1 (20)

119906 minus 1198784(119906 120593) = 119874 (ℎ

4) (21)

Let 120596ℎ119899

119868sub 120596

ℎ119899 be the set of such points 119875 isin 120596ℎ119899

for which all points 119875119897in expression (19) are in cup

119872

119896=1Π

ℎ

119896 If

some points 119875119897in (19) emerge through the side 120574

119898 then the

set of such points 119875 is denoted by 120596ℎ119899

119863 According to the

construction of 1198784 in [14] the expression 1198784(119906

ℎ 120593) at each

point 119875 isin 120596ℎ119899

= 120596ℎ119899

119868cup 120596

ℎ119899

119863can be expressed as follows

1198784(119906

ℎ 120593)

=

1198784119906ℎ 119875 isin 120596

ℎ119899

119868

1198784(119906

ℎminus

3

sum

119896=0

119886119896Re 119911119896) + (

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(22)

where

119886119896=

1

119896

119889119896120593119898(119904)

119889119904119896

100381610038161003816100381610038161003816100381610038161003816119904=119904119875

119896 = 0 1 2 3 (23)

and 119904119875corresponds to such a point119876 isin 120574

119898for which the line

119875119876 is perpendicular to 120574119898

Let

119876119895= 119876

119895(119903

119895 120579

119895) 119876

119902

1198952= 119876

119895(119903

1198952 120579

119902

119895) (24)

The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876

120576

119895and 119876

119902120576

1198952of the

quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876

120576

119895minus 119876

119895

10038161003816100381610038161003816le 119888

1120576

10038161003816100381610038161003816119876

119902120576

1198952minus 119876

119902

1198952

10038161003816100381610038161003816le 119888

2120576 (25)

where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888

2are constants

independent of 120576Consider the system of linear algebraic equations

119906120576

ℎ= 119861119906

120576

ℎon Π

ℎ

119896

119906120576

ℎ= 120593

119898on 120578

ℎ

1198961cap 120574

119898

119906120576

ℎ(119903

119895 120579

119895)

= 119876120576

119895+ 120573

119895

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

times 119877119895(119903

119895 120579

119895 120579

119902

119895) on (119903

119895 120579

119895) isin 119905

ℎ

119896119895

119906120576

ℎ= 119878

4119906120576

ℎon 120596

ℎ119899

(26)

where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864

Abstract and Applied Analysis 5

Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2

minus984858 and 120576 = 5 times 10minus13

(2minus984858 119899) 120577

120576

ℎ119866119873119878 R

984858

119866119873119878

(2minus4 60) 1609 times 10

minus815577

(2minus5 170) 1033 times 10

minus9

(2minus5 130) 1191 times 10

minus916690

(2minus6 150) 7136 times 10

minus11

(2minus5 140) 1136 times 10

minus916259

(2minus6 170) 6991 times 10

minus11

(2minus6 100) 2169 times 10

minus1017096

(2minus7 130) 1269 times 10

minus11

Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =

2minus984858 and 120576 = 5 times 10

minus13

(2minus984858 119899) 120577

120576

ℎ119866119878 R

984858

119866119878

(2minus4 100) 1931 times 10

minus816078

(2minus5 150) 1182 times 10

minus9

(2minus5 130) 1294 times 10

minus916789

(2minus6 150) 7708 times 10

minus11

(2minus5 140) 1312 times 10

minus917967

(2minus6 170) 7304 times 10

minus11

(2minus6 100) 2389 times 10

minus1018164

(2minus7 130) 1315 times 10

minus11

Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) 120577

120576

ℎ119866119873119878 120577

120576

ℎ119866119878 Iteration

(16 70) 1139 times 10minus8

1572 times 10minus8

22

(32 170) 1033 times 10minus9

1184 times 10minus9

23

(64 170) 6990 times 10minus11

7304 times 10minus11

24

(128 200) 8628 times 10minus12

8833 times 10minus12

25

Let 119879lowast

119895= 119879

119895(119903

lowast

119895) be the sector where 119903lowast

119895= (119903

1198952+119903

1198953)2 119895 isin

119864 and let 119906120576ℎ(119903

1198952 120579

119902

119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution

values of the system (26) on 119881ℎ

119895(at the quadrature nodes)

The function

119880120576

ℎ(119903

119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ 120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) (119906

120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

(27)

defined on 119879lowast

119895 is called an approximate solution of the

problem (1) on the closed block 1198793

119895 119895 isin 119864

Definition 5 The system (26) and (27) is called the system ofblock-grid equations

Theorem 6 There is a natural number 1198990 such that for all

119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution

Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899

0and 120590 such that for all 119899 ge 119899

0

max(119903119895120579119895)isin119879

3

119895

120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) le 120590 lt 1 (28)

The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)

Theorem 7 There exists a natural number 1198990 such that for all

119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)

where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid

max119866ℎ119899

119879

1003816100381610038161003816119906120576

ℎminus 119906

1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816le 119888

119901(ℎ

4+ 120576) on 119879

3

119895

(31)

for integer 1120572119895when 119901 ge 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895

(32)

for any 1120572119895 if 0 le 119901 lt 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895 119860

119895

(33)

for noninteger 1120572119895 when 119901 gt 1120572

119895 Everywhere 0 le 119902 le 119901 119906

is the exact solution of the problem (1) and119880120576

ℎ(119903

119895 120579

119895) is defined

by formula (27)

Proof Let

120585120576

ℎ= 119906

120576

ℎminus 119906 (34)

where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866

ℎ119899

119879

of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576

ℎsatisfies the system of difference equations

120585120576

ℎ= 119861120585

120576

ℎ+ 119903

1

ℎon Π

ℎ

119896

120585120576

ℎ= 0 on 120578

ℎ

1198961

120585120576

ℎ(119903

119895 120579

119895) = 120573

119895

119899(119895)

sum

119902=1

120585120576

ℎ(119903

1198952 120579

119902

119895) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

+ 1199032

119895ℎ (119903

119895 120579

119895) isin 119905

ℎ

119896119895

120585120576

ℎ= 119878

4120585120576

ℎ+ 119903

3

ℎon 120596

ℎ119899

(35)

6 Abstract and Applied Analysis

Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 3895 times 10minus7

3895 times 10minus7

(32 170) 4627 times 10minus8

4627 times 10minus8

(64 170) 1124 times 10minus9

3125 times 10minus9

(128 200) 2214 times 10minus10

2233 times 10minus10

Table 5 In119866119878 the minimum errors of the derivatives over the pairs

(ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 9663 times 10minus6

9663 times 10minus6

(32 170) 9653 times 10minus6

9653 times 10minus6

(64 170) 9649 times 10minus6

9649 times 10minus6

(128 200) 9648 times 10minus6

9648 times 10minus6

where 1 le 119896 le 119872 119895 isin 119864

1199031

ℎ= 119861119906 minus 119906 on cup

119872

119896=1Π

ℎ

119896 (36)

1199032

119895ℎ= 120573

119895

119899(119895)

sum

119902=1

(119906 (1199031198952 120579

119902

119895) minus 119876

119902120576

1198952) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

minus (119906 minus 119876120576

119895) on cup

119872

119896=1(cup

119895isin119864119905ℎ

119896119895)

(37)

1199033

ℎ

=

1198784119906 minus 119906 on 120596

ℎ119899

119868

1198784(119906 minus

3

sum

119896=0

119886119896Re 119911119896) minus (119906 minus

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(38)

On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows

The function 119880120576

ℎ(119903

119895 120579

119895) given by formula (27) defined

on the closed sector 119879lowast

119895 119895 isin 119864 where 119903

lowast

119895= (119903

1198952+ 119903

1198953)2

and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879

2

119895 119881

119895 119895 isin 119864 and then the

difference function 120577120576

ℎ(119903

119895 120579

119895) = 119880

120576

ℎ(119903

119895 120579

119895)minus119906(119903

119895 120579

119895) is defined

on 119879lowast

119895 119895 isin 119864 and

120577120576

ℎ(119903

lowast

119895 0) = 120577

120576

ℎ(119903

lowast

119895 120572

119895120587) = 0 119895 isin 119864 (39)

On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge

max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain

10038161003816100381610038161003816120577120576

ℎ(119903

119895 120579

119895)10038161003816100381610038161003816le 119888 (ℎ

4+ 120576) on 119879

lowast

119895 119895 isin 119864 (40)

Furthermore the function 120577120576

ℎ(119903

119895 120579

119895) continuous on 119879

lowast

119895is a

solution of the following Dirichlet problem

Δ120577120576

ℎ= 0 on 119879

lowast

119895

120577120576

ℎ= 0 on 120574

119898cap 119879

lowast

119895 119898 = 119895 minus 1 119895

120577120576

ℎ(119903

lowast

119895 120579

119895) = 119880

120576

ℎ(119903

lowast

119895 120579

119895) minus 119906 (119903

lowast

119895 120579

119895) 0 le 120579

119895le 120572

119895120587

(41)

Since 1198793

119895sub 119879

lowast

119895 on the basis of (39) and (40) from Lemma

612 in [16] inequalities (31)ndash(33) of Theorem 7 follow

5 Stress Intensity Factor

Let in the condition 120593119895isin 119862

4120582(120574

119895) the exponent 120582 be such

that

120572119895(4 + 120582) = 0 2120572

119895(4 + 120582) = 0 (42)

where sdot is the symbol of fractional partThese conditions forthe given 120572

119895can be fulfilled by decreasing 120582

On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879

lowast

119895 119895 isin 119864 as follows

119906 (119909119895 119910

119895) = (119909

119895 119910

119895) +

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911

+

119899120572119895

sum

119896=1

120591(119895)

119896119903119896120572119895

119895sin

119896120579119895

120572119895

(43)

where 119899120572119895

= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909

119895+119894119910

119895 120583(119895)

119896

and 120591(119895)

119896are some numbers and (119909

119895 119910

119895) isin 119862

4120582(119879

2

119895) is the

harmonic on 1198792

119895 By taking 120579

119895= 120572

1198951205872 from the formula

(43) it follows that the coefficient 120591(119895)1

which is called the stressintensity factor can be represented as

120591(119895)

1= lim

119903119895rarr0

1

1199031120572119895

119895

(119906 (119909119895 119910

119895) minus (119909

119895 119910

119895)

minus

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911)

(44)

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

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Function Spaces

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

minus1 minus05 0 05 1minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Π1

Π212057810

12057820

119886

119887119888

119889119890

120579

1198601 1205741

1205740

11987941

11987931

11987921

11987911

Π3

Π4

12057820

12057830

12057830

12057840

119903

11990311

11990312

11990313 11990314

119866119878

Figure 2 Description of BGM for the L-shaped domain

the curvilinear part of the boundary of the sector 1198792

119895 and let

119905119896119895

= 120578119896cap 119879

3

119895 We choose a natural number 119899 and define

the quantities 119899(119895) = max4 [120572119895119899] 120573

119895= 120572

119895120587119899(119895) and

120579119898

119895= (119898 minus 12)120573

119895 119895 isin 119864 1 le 119898 le 119899(119895) On the arc 119881

119895

we take the points (1199031198952 120579

119898

119895) 1 le 119898 le 119899(119895) and denote the

set of these points by 119881119899

119895 We introduce the parameter ℎ isin

(0 12060004] where 120600

0is a gluing depth of the rectanglesΠ

119896 119896 =

1 2 119872 and define a square grid on Π119896 119896 = 1 2 119872

withmaximal possible step ℎ119896le minℎmin119886

1119896 119886

21198966 such

that the boundary 120578119896lies entirely on the grid lines Let Πℎ

119896be

the set of grid nodes on Π119896 let 120578ℎ

119896be the set of nodes on 120578

119896

and let Πℎ

119896= Π

ℎ

119896cup 120578

ℎ

119896 We denote the set of nodes on the

closure of 120578119896cap 119866

119879by 120578

ℎ

1198960 the set of nodes on 119905

119896119895by 119905

ℎ

119896119895 and

the set of remaining nodes on 120578119896by 120578

ℎ

1198961

Let

120596ℎ119899

= (cup119872

119896=1120578ℎ

1198960) cup (cup

119895isin119864119881

119899

119895)

119866ℎ119899

119879= 120596

ℎ119899cup (cup

119872

119896=1Π

ℎ

119896)

(18)

Let 120593 = 120593119895119873

119895=1 where 120593

119895isin 119862

4120582(120574

119895) 0 lt 120582 lt 1 is

the given function in (1) We use the matching operator 1198784

at the points of the set 120596ℎ119899 constructed in [14] The value of1198784(119906

ℎ 120593) at the point 119875 isin 120596

ℎ119899 is expressed linearly in termsof the values of 119906

ℎat the points 119875

119896of the grid constructed on

Π119896(119875)

(119875 isin Π119896(119875)

) some part of whose boundary located in119866 is the maximum distance away from 119875 and in terms of theboundary values of 120593(119898)

119898 = 0 1 2 3 at a fixed number ofpoints Moreover 1198784(119906

ℎ 0) has the representation

1198784(119906

ℎ 0) = sum

0le119897le15

120585119897119906ℎ119897 (19)

where 119906ℎ119896

= 119906ℎ(119875

119896)

120585119897ge 0 sum

0le119897le15

120585119897= 1 (20)

119906 minus 1198784(119906 120593) = 119874 (ℎ

4) (21)

Let 120596ℎ119899

119868sub 120596

ℎ119899 be the set of such points 119875 isin 120596ℎ119899

for which all points 119875119897in expression (19) are in cup

119872

119896=1Π

ℎ

119896 If

some points 119875119897in (19) emerge through the side 120574

119898 then the

set of such points 119875 is denoted by 120596ℎ119899

119863 According to the

construction of 1198784 in [14] the expression 1198784(119906

ℎ 120593) at each

point 119875 isin 120596ℎ119899

= 120596ℎ119899

119868cup 120596

ℎ119899

119863can be expressed as follows

1198784(119906

ℎ 120593)

=

1198784119906ℎ 119875 isin 120596

ℎ119899

119868

1198784(119906

ℎminus

3

sum

119896=0

119886119896Re 119911119896) + (

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(22)

where

119886119896=

1

119896

119889119896120593119898(119904)

119889119904119896

100381610038161003816100381610038161003816100381610038161003816119904=119904119875

119896 = 0 1 2 3 (23)

and 119904119875corresponds to such a point119876 isin 120574

119898for which the line

119875119876 is perpendicular to 120574119898

Let

119876119895= 119876

119895(119903

119895 120579

119895) 119876

119902

1198952= 119876

119895(119903

1198952 120579

119902

119895) (24)

The quantities in (24) are given by (4) and (5) which containintegrals that have not been computed exactly in the generalcase Assume that approximate values 119876

120576

119895and 119876

119902120576

1198952of the

quantities in (24) are known with accuracy 120576 gt 0 that is10038161003816100381610038161003816119876

120576

119895minus 119876

119895

10038161003816100381610038161003816le 119888

1120576

10038161003816100381610038161003816119876

119902120576

1198952minus 119876

119902

1198952

10038161003816100381610038161003816le 119888

2120576 (25)

where 119895 isin 119864 1 le 119902 le 119899(119895) and 1198881 119888

2are constants

independent of 120576Consider the system of linear algebraic equations

119906120576

ℎ= 119861119906

120576

ℎon Π

ℎ

119896

119906120576

ℎ= 120593

119898on 120578

ℎ

1198961cap 120574

119898

119906120576

ℎ(119903

119895 120579

119895)

= 119876120576

119895+ 120573

119895

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

times 119877119895(119903

119895 120579

119895 120579

119902

119895) on (119903

119895 120579

119895) isin 119905

ℎ

119896119895

119906120576

ℎ= 119878

4119906120576

ℎon 120596

ℎ119899

(26)

where 1 le 119896 le 119872 1 le 119898 le 119873 and 119895 isin 119864

Abstract and Applied Analysis 5

Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2

minus984858 and 120576 = 5 times 10minus13

(2minus984858 119899) 120577

120576

ℎ119866119873119878 R

984858

119866119873119878

(2minus4 60) 1609 times 10

minus815577

(2minus5 170) 1033 times 10

minus9

(2minus5 130) 1191 times 10

minus916690

(2minus6 150) 7136 times 10

minus11

(2minus5 140) 1136 times 10

minus916259

(2minus6 170) 6991 times 10

minus11

(2minus6 100) 2169 times 10

minus1017096

(2minus7 130) 1269 times 10

minus11

Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =

2minus984858 and 120576 = 5 times 10

minus13

(2minus984858 119899) 120577

120576

ℎ119866119878 R

984858

119866119878

(2minus4 100) 1931 times 10

minus816078

(2minus5 150) 1182 times 10

minus9

(2minus5 130) 1294 times 10

minus916789

(2minus6 150) 7708 times 10

minus11

(2minus5 140) 1312 times 10

minus917967

(2minus6 170) 7304 times 10

minus11

(2minus6 100) 2389 times 10

minus1018164

(2minus7 130) 1315 times 10

minus11

Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) 120577

120576

ℎ119866119873119878 120577

120576

ℎ119866119878 Iteration

(16 70) 1139 times 10minus8

1572 times 10minus8

22

(32 170) 1033 times 10minus9

1184 times 10minus9

23

(64 170) 6990 times 10minus11

7304 times 10minus11

24

(128 200) 8628 times 10minus12

8833 times 10minus12

25

Let 119879lowast

119895= 119879

119895(119903

lowast

119895) be the sector where 119903lowast

119895= (119903

1198952+119903

1198953)2 119895 isin

119864 and let 119906120576ℎ(119903

1198952 120579

119902

119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution

values of the system (26) on 119881ℎ

119895(at the quadrature nodes)

The function

119880120576

ℎ(119903

119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ 120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) (119906

120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

(27)

defined on 119879lowast

119895 is called an approximate solution of the

problem (1) on the closed block 1198793

119895 119895 isin 119864

Definition 5 The system (26) and (27) is called the system ofblock-grid equations

Theorem 6 There is a natural number 1198990 such that for all

119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution

Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899

0and 120590 such that for all 119899 ge 119899

0

max(119903119895120579119895)isin119879

3

119895

120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) le 120590 lt 1 (28)

The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)

Theorem 7 There exists a natural number 1198990 such that for all

119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)

where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid

max119866ℎ119899

119879

1003816100381610038161003816119906120576

ℎminus 119906

1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816le 119888

119901(ℎ

4+ 120576) on 119879

3

119895

(31)

for integer 1120572119895when 119901 ge 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895

(32)

for any 1120572119895 if 0 le 119901 lt 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895 119860

119895

(33)

for noninteger 1120572119895 when 119901 gt 1120572

119895 Everywhere 0 le 119902 le 119901 119906

is the exact solution of the problem (1) and119880120576

ℎ(119903

119895 120579

119895) is defined

by formula (27)

Proof Let

120585120576

ℎ= 119906

120576

ℎminus 119906 (34)

where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866

ℎ119899

119879

of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576

ℎsatisfies the system of difference equations

120585120576

ℎ= 119861120585

120576

ℎ+ 119903

1

ℎon Π

ℎ

119896

120585120576

ℎ= 0 on 120578

ℎ

1198961

120585120576

ℎ(119903

119895 120579

119895) = 120573

119895

119899(119895)

sum

119902=1

120585120576

ℎ(119903

1198952 120579

119902

119895) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

+ 1199032

119895ℎ (119903

119895 120579

119895) isin 119905

ℎ

119896119895

120585120576

ℎ= 119878

4120585120576

ℎ+ 119903

3

ℎon 120596

ℎ119899

(35)

6 Abstract and Applied Analysis

Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 3895 times 10minus7

3895 times 10minus7

(32 170) 4627 times 10minus8

4627 times 10minus8

(64 170) 1124 times 10minus9

3125 times 10minus9

(128 200) 2214 times 10minus10

2233 times 10minus10

Table 5 In119866119878 the minimum errors of the derivatives over the pairs

(ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 9663 times 10minus6

9663 times 10minus6

(32 170) 9653 times 10minus6

9653 times 10minus6

(64 170) 9649 times 10minus6

9649 times 10minus6

(128 200) 9648 times 10minus6

9648 times 10minus6

where 1 le 119896 le 119872 119895 isin 119864

1199031

ℎ= 119861119906 minus 119906 on cup

119872

119896=1Π

ℎ

119896 (36)

1199032

119895ℎ= 120573

119895

119899(119895)

sum

119902=1

(119906 (1199031198952 120579

119902

119895) minus 119876

119902120576

1198952) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

minus (119906 minus 119876120576

119895) on cup

119872

119896=1(cup

119895isin119864119905ℎ

119896119895)

(37)

1199033

ℎ

=

1198784119906 minus 119906 on 120596

ℎ119899

119868

1198784(119906 minus

3

sum

119896=0

119886119896Re 119911119896) minus (119906 minus

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(38)

On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows

The function 119880120576

ℎ(119903

119895 120579

119895) given by formula (27) defined

on the closed sector 119879lowast

119895 119895 isin 119864 where 119903

lowast

119895= (119903

1198952+ 119903

1198953)2

and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879

2

119895 119881

119895 119895 isin 119864 and then the

difference function 120577120576

ℎ(119903

119895 120579

119895) = 119880

120576

ℎ(119903

119895 120579

119895)minus119906(119903

119895 120579

119895) is defined

on 119879lowast

119895 119895 isin 119864 and

120577120576

ℎ(119903

lowast

119895 0) = 120577

120576

ℎ(119903

lowast

119895 120572

119895120587) = 0 119895 isin 119864 (39)

On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge

max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain

10038161003816100381610038161003816120577120576

ℎ(119903

119895 120579

119895)10038161003816100381610038161003816le 119888 (ℎ

4+ 120576) on 119879

lowast

119895 119895 isin 119864 (40)

Furthermore the function 120577120576

ℎ(119903

119895 120579

119895) continuous on 119879

lowast

119895is a

solution of the following Dirichlet problem

Δ120577120576

ℎ= 0 on 119879

lowast

119895

120577120576

ℎ= 0 on 120574

119898cap 119879

lowast

119895 119898 = 119895 minus 1 119895

120577120576

ℎ(119903

lowast

119895 120579

119895) = 119880

120576

ℎ(119903

lowast

119895 120579

119895) minus 119906 (119903

lowast

119895 120579

119895) 0 le 120579

119895le 120572

119895120587

(41)

Since 1198793

119895sub 119879

lowast

119895 on the basis of (39) and (40) from Lemma

612 in [16] inequalities (31)ndash(33) of Theorem 7 follow

5 Stress Intensity Factor

Let in the condition 120593119895isin 119862

4120582(120574

119895) the exponent 120582 be such

that

120572119895(4 + 120582) = 0 2120572

119895(4 + 120582) = 0 (42)

where sdot is the symbol of fractional partThese conditions forthe given 120572

119895can be fulfilled by decreasing 120582

On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879

lowast

119895 119895 isin 119864 as follows

119906 (119909119895 119910

119895) = (119909

119895 119910

119895) +

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911

+

119899120572119895

sum

119896=1

120591(119895)

119896119903119896120572119895

119895sin

119896120579119895

120572119895

(43)

where 119899120572119895

= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909

119895+119894119910

119895 120583(119895)

119896

and 120591(119895)

119896are some numbers and (119909

119895 119910

119895) isin 119862

4120582(119879

2

119895) is the

harmonic on 1198792

119895 By taking 120579

119895= 120572

1198951205872 from the formula

(43) it follows that the coefficient 120591(119895)1

which is called the stressintensity factor can be represented as

120591(119895)

1= lim

119903119895rarr0

1

1199031120572119895

119895

(119906 (119909119895 119910

119895) minus (119909

119895 119910

119895)

minus

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911)

(44)

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Table 1 The order of convergence in the ldquononsingularrdquo part whenℎ = 2

minus984858 and 120576 = 5 times 10minus13

(2minus984858 119899) 120577

120576

ℎ119866119873119878 R

984858

119866119873119878

(2minus4 60) 1609 times 10

minus815577

(2minus5 170) 1033 times 10

minus9

(2minus5 130) 1191 times 10

minus916690

(2minus6 150) 7136 times 10

minus11

(2minus5 140) 1136 times 10

minus916259

(2minus6 170) 6991 times 10

minus11

(2minus6 100) 2169 times 10

minus1017096

(2minus7 130) 1269 times 10

minus11

Table 2 The order of convergence in the ldquosingularrdquo part when ℎ =

2minus984858 and 120576 = 5 times 10

minus13

(2minus984858 119899) 120577

120576

ℎ119866119878 R

984858

119866119878

(2minus4 100) 1931 times 10

minus816078

(2minus5 150) 1182 times 10

minus9

(2minus5 130) 1294 times 10

minus916789

(2minus6 150) 7708 times 10

minus11

(2minus5 140) 1312 times 10

minus917967

(2minus6 170) 7304 times 10

minus11

(2minus6 100) 2389 times 10

minus1018164

(2minus7 130) 1315 times 10

minus11

Table 3 The minimum errors of the solution over the pairs (ℎminus1 119899)in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) 120577

120576

ℎ119866119873119878 120577

120576

ℎ119866119878 Iteration

(16 70) 1139 times 10minus8

1572 times 10minus8

22

(32 170) 1033 times 10minus9

1184 times 10minus9

23

(64 170) 6990 times 10minus11

7304 times 10minus11

24

(128 200) 8628 times 10minus12

8833 times 10minus12

25

Let 119879lowast

119895= 119879

119895(119903

lowast

119895) be the sector where 119903lowast

119895= (119903

1198952+119903

1198953)2 119895 isin

119864 and let 119906120576ℎ(119903

1198952 120579

119902

119895) 1 le 119902 le 119899(119895) 119895 isin 119864 be the solution

values of the system (26) on 119881ℎ

119895(at the quadrature nodes)

The function

119880120576

ℎ(119903

119895 120579

119895) = 119876

119895(119903

119895 120579

119895)

+ 120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) (119906

120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952)

(27)

defined on 119879lowast

119895 is called an approximate solution of the

problem (1) on the closed block 1198793

119895 119895 isin 119864

Definition 5 The system (26) and (27) is called the system ofblock-grid equations

Theorem 6 There is a natural number 1198990 such that for all

119899 ge 1198990and for any 120576 gt 0 the system (26) has a unique solution

Proof From the estimation (229) in [15] follows the existenceof the positive constants 119899

0and 120590 such that for all 119899 ge 119899

0

max(119903119895120579119895)isin119879

3

119895

120573119895

119899(119895)

sum

119902=1

119877119895(119903

119895 120579

119895 120579

119902

119895) le 120590 lt 1 (28)

The proof is obtained on the basis of principle of maximumby taking into account (14) (19) (20) and (28)

Theorem 7 There exists a natural number 1198990 such that for all

119899 ge max 1198990 [ln1+120600ℎminus1] + 1 (29)

where 120600 gt 0 is a fixed number and for any 120576 gt 0 the followinginequalities are valid

max119866ℎ119899

119879

1003816100381610038161003816119906120576

ℎminus 119906

1003816100381610038161003816 le 119888 (ℎ4+ 120576) (30)

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816le 119888

119901(ℎ

4+ 120576) on 119879

3

119895

(31)

for integer 1120572119895when 119901 ge 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895

(32)

for any 1120572119895 if 0 le 119901 lt 1120572

119895

10038161003816100381610038161003816100381610038161003816

120597119901

120597119909119901minus119902120597119910119902(119880

120576

ℎ(119903

119895 120579

119895) minus 119906 (119903

119895 120579

119895))

10038161003816100381610038161003816100381610038161003816

le119888119901(ℎ

4+ 120576)

119903119901minus1120572119895

on 1198793

119895 119860

119895

(33)

for noninteger 1120572119895 when 119901 gt 1120572

119895 Everywhere 0 le 119902 le 119901 119906

is the exact solution of the problem (1) and119880120576

ℎ(119903

119895 120579

119895) is defined

by formula (27)

Proof Let

120585120576

ℎ= 119906

120576

ℎminus 119906 (34)

where 119906120576ℎis a solution of system (26) and 119906 is the trace on119866

ℎ119899

119879

of the solution of (1) On the basis of (1) (26) and (34) theerror 120585120576

ℎsatisfies the system of difference equations

120585120576

ℎ= 119861120585

120576

ℎ+ 119903

1

ℎon Π

ℎ

119896

120585120576

ℎ= 0 on 120578

ℎ

1198961

120585120576

ℎ(119903

119895 120579

119895) = 120573

119895

119899(119895)

sum

119902=1

120585120576

ℎ(119903

1198952 120579

119902

119895) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

+ 1199032

119895ℎ (119903

119895 120579

119895) isin 119905

ℎ

119896119895

120585120576

ℎ= 119878

4120585120576

ℎ+ 119903

3

ℎon 120596

ℎ119899

(35)

6 Abstract and Applied Analysis

Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 3895 times 10minus7

3895 times 10minus7

(32 170) 4627 times 10minus8

4627 times 10minus8

(64 170) 1124 times 10minus9

3125 times 10minus9

(128 200) 2214 times 10minus10

2233 times 10minus10

Table 5 In119866119878 the minimum errors of the derivatives over the pairs

(ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 9663 times 10minus6

9663 times 10minus6

(32 170) 9653 times 10minus6

9653 times 10minus6

(64 170) 9649 times 10minus6

9649 times 10minus6

(128 200) 9648 times 10minus6

9648 times 10minus6

where 1 le 119896 le 119872 119895 isin 119864

1199031

ℎ= 119861119906 minus 119906 on cup

119872

119896=1Π

ℎ

119896 (36)

1199032

119895ℎ= 120573

119895

119899(119895)

sum

119902=1

(119906 (1199031198952 120579

119902

119895) minus 119876

119902120576

1198952) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

minus (119906 minus 119876120576

119895) on cup

119872

119896=1(cup

119895isin119864119905ℎ

119896119895)

(37)

1199033

ℎ

=

1198784119906 minus 119906 on 120596

ℎ119899

119868

1198784(119906 minus

3

sum

119896=0

119886119896Re 119911119896) minus (119906 minus

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(38)

On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows

The function 119880120576

ℎ(119903

119895 120579

119895) given by formula (27) defined

on the closed sector 119879lowast

119895 119895 isin 119864 where 119903

lowast

119895= (119903

1198952+ 119903

1198953)2

and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879

2

119895 119881

119895 119895 isin 119864 and then the

difference function 120577120576

ℎ(119903

119895 120579

119895) = 119880

120576

ℎ(119903

119895 120579

119895)minus119906(119903

119895 120579

119895) is defined

on 119879lowast

119895 119895 isin 119864 and

120577120576

ℎ(119903

lowast

119895 0) = 120577

120576

ℎ(119903

lowast

119895 120572

119895120587) = 0 119895 isin 119864 (39)

On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge

max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain

10038161003816100381610038161003816120577120576

ℎ(119903

119895 120579

119895)10038161003816100381610038161003816le 119888 (ℎ

4+ 120576) on 119879

lowast

119895 119895 isin 119864 (40)

Furthermore the function 120577120576

ℎ(119903

119895 120579

119895) continuous on 119879

lowast

119895is a

solution of the following Dirichlet problem

Δ120577120576

ℎ= 0 on 119879

lowast

119895

120577120576

ℎ= 0 on 120574

119898cap 119879

lowast

119895 119898 = 119895 minus 1 119895

120577120576

ℎ(119903

lowast

119895 120579

119895) = 119880

120576

ℎ(119903

lowast

119895 120579

119895) minus 119906 (119903

lowast

119895 120579

119895) 0 le 120579

119895le 120572

119895120587

(41)

Since 1198793

119895sub 119879

lowast

119895 on the basis of (39) and (40) from Lemma

612 in [16] inequalities (31)ndash(33) of Theorem 7 follow

5 Stress Intensity Factor

Let in the condition 120593119895isin 119862

4120582(120574

119895) the exponent 120582 be such

that

120572119895(4 + 120582) = 0 2120572

119895(4 + 120582) = 0 (42)

where sdot is the symbol of fractional partThese conditions forthe given 120572

119895can be fulfilled by decreasing 120582

On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879

lowast

119895 119895 isin 119864 as follows

119906 (119909119895 119910

119895) = (119909

119895 119910

119895) +

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911

+

119899120572119895

sum

119896=1

120591(119895)

119896119903119896120572119895

119895sin

119896120579119895

120572119895

(43)

where 119899120572119895

= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909

119895+119894119910

119895 120583(119895)

119896

and 120591(119895)

119896are some numbers and (119909

119895 119910

119895) isin 119862

4120582(119879

2

119895) is the

harmonic on 1198792

119895 By taking 120579

119895= 120572

1198951205872 from the formula

(43) it follows that the coefficient 120591(119895)1

which is called the stressintensity factor can be represented as

120591(119895)

1= lim

119903119895rarr0

1

1199031120572119895

119895

(119906 (119909119895 119910

119895) minus (119909

119895 119910

119895)

minus

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911)

(44)

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Table 4 In 119866119878cap 119903 ge 02 the minimum errors of the derivatives over the pairs (ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878cap 119903ge02

11990313

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 3895 times 10minus7

3895 times 10minus7

(32 170) 4627 times 10minus8

4627 times 10minus8

(64 170) 1124 times 10minus9

3125 times 10minus9

(128 200) 2214 times 10minus10

2233 times 10minus10

Table 5 In119866119878 the minimum errors of the derivatives over the pairs

(ℎminus1 119899) in maximum norm when 120576 = 5 times 10

minus13

(ℎminus1 119899) Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119909minus

120597119906

120597119909

10038171003817100381710038171003817100381710038171003817Max

119866119878119903

13

10038171003817100381710038171003817100381710038171003817

120597119880120576

ℎ

120597119910minus

120597119906

120597119910

10038171003817100381710038171003817100381710038171003817

(16 70) 9663 times 10minus6

9663 times 10minus6

(32 170) 9653 times 10minus6

9653 times 10minus6

(64 170) 9649 times 10minus6

9649 times 10minus6

(128 200) 9648 times 10minus6

9648 times 10minus6

where 1 le 119896 le 119872 119895 isin 119864

1199031

ℎ= 119861119906 minus 119906 on cup

119872

119896=1Π

ℎ

119896 (36)

1199032

119895ℎ= 120573

119895

119899(119895)

sum

119902=1

(119906 (1199031198952 120579

119902

119895) minus 119876

119902120576

1198952) 119877

119895(119903

119895 120579

119895 120579

119902

119895)

minus (119906 minus 119876120576

119895) on cup

119872

119896=1(cup

119895isin119864119905ℎ

119896119895)

(37)

1199033

ℎ

=

1198784119906 minus 119906 on 120596

ℎ119899

119868

1198784(119906 minus

3

sum

119896=0

119886119896Re 119911119896) minus (119906 minus

3

sum

119896=0

119886119896Re 119911119896)

119875

119875 isin 120596ℎ119899

119863

(38)

On the basis of estimations (15) (21) (25) and Lemma 1by analogy to the proof of Theorem 43 in [9] the proof ofinequality (30) follows

The function 119880120576

ℎ(119903

119895 120579

119895) given by formula (27) defined

on the closed sector 119879lowast

119895 119895 isin 119864 where 119903

lowast

119895= (119903

1198952+ 119903

1198953)2

and the integral representation (8) of the exact solution ofthe problem (1) is given on 119879

2

119895 119881

119895 119895 isin 119864 and then the

difference function 120577120576

ℎ(119903

119895 120579

119895) = 119880

120576

ℎ(119903

119895 120579

119895)minus119906(119903

119895 120579

119895) is defined

on 119879lowast

119895 119895 isin 119864 and

120577120576

ℎ(119903

lowast

119895 0) = 120577

120576

ℎ(119903

lowast

119895 120572

119895120587) = 0 119895 isin 119864 (39)

On the basis of Lemma 611 from [16] (25) and (28) for 119899 ge

max1198990 [ln1+120600ℎminus1]+1120600 gt 0 is a fixed number andwe obtain

10038161003816100381610038161003816120577120576

ℎ(119903

119895 120579

119895)10038161003816100381610038161003816le 119888 (ℎ

4+ 120576) on 119879

lowast

119895 119895 isin 119864 (40)

Furthermore the function 120577120576

ℎ(119903

119895 120579

119895) continuous on 119879

lowast

119895is a

solution of the following Dirichlet problem

Δ120577120576

ℎ= 0 on 119879

lowast

119895

120577120576

ℎ= 0 on 120574

119898cap 119879

lowast

119895 119898 = 119895 minus 1 119895

120577120576

ℎ(119903

lowast

119895 120579

119895) = 119880

120576

ℎ(119903

lowast

119895 120579

119895) minus 119906 (119903

lowast

119895 120579

119895) 0 le 120579

119895le 120572

119895120587

(41)

Since 1198793

119895sub 119879

lowast

119895 on the basis of (39) and (40) from Lemma

612 in [16] inequalities (31)ndash(33) of Theorem 7 follow

5 Stress Intensity Factor

Let in the condition 120593119895isin 119862

4120582(120574

119895) the exponent 120582 be such

that

120572119895(4 + 120582) = 0 2120572

119895(4 + 120582) = 0 (42)

where sdot is the symbol of fractional partThese conditions forthe given 120572

119895can be fulfilled by decreasing 120582

On the basis of Section 2 of [11] a solution of the problem(1) can be represented in 119879

lowast

119895 119895 isin 119864 as follows

119906 (119909119895 119910

119895) = (119909

119895 119910

119895) +

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911

+

119899120572119895

sum

119896=1

120591(119895)

119896119903119896120572119895

119895sin

119896120579119895

120572119895

(43)

where 119899120572119895

= [120572119895(4+120582)] [sdot] is the integer part 119911 = 119909

119895+119894119910

119895 120583(119895)

119896

and 120591(119895)

119896are some numbers and (119909

119895 119910

119895) isin 119862

4120582(119879

2

119895) is the

harmonic on 1198792

119895 By taking 120579

119895= 120572

1198951205872 from the formula

(43) it follows that the coefficient 120591(119895)1

which is called the stressintensity factor can be represented as

120591(119895)

1= lim

119903119895rarr0

1

1199031120572119895

119895

(119906 (119909119895 119910

119895) minus (119909

119895 119910

119895)

minus

4

sum

119896=0

120583(119895)

119896Im 119911

119896 ln 119911)

(44)

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Table 6 The stress intensity factor 1205911205981119899

for 119899 = 70 170 200 when 120576 = 5 times 10minus13

ℎminus1

120591120576

170120591120576

1170120591120576

1200

16 1000000014856688 1000000017180415 1000000017197438

32 1000000005800844 1000000001230267 1000000001236709

64 1000000004138169 1000000000073107 1000000000079938

128 1000000004053153 1000000000003531 1000000000003267

10minus15 10minus10 10minus5 10010minus10

10minus5

100

120576

Max

erro

r

ℎ = 116 119899 = 70

(a)

10minus10

10minus5

100

10minus15 10minus10 10minus5 100

120576

Max

erro

r

ℎ = 132 119899 = 170

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 3 Dependence on 120576 for ℎminus1 = 16 32

From formula (44) it follows that 120591(119895)

1can be approxi-

mated by

120591(119895)120576

1119899

= lim119903119895rarr0

1

1199031120572119895

119895

(119880120576

ℎ(119903

119895 120579

119895)

minus(120593119895(119904

119895) + (120593

119895minus1(119904

119895) minus 120593

119895(119904

119895))

120579119895

120572119895120587))

(45)

Using formula (3) (4) and (27) from (45) for the stressintensity factor (see [17]) we obtain the next formula

120591(119895)120576

1119899=

1

120587int

1205901198950

0

1205931198950(119905

120572119895) 119889119905

1199052+

1

120587int

1205901198951

0

1205931198951(119905

120572119895) 119889119905

1199052

+2

119899 (119895) 1199031120572119895

1198952

119899(119895)

sum

119902=1

(119906120576

ℎ(119903

1198952 120579

119902

119895) minus 119876

119902120576

1198952) sin 1

120572119895

120579119902

119895

(46)

This formula is obtained for the second-order BGM in [8]

10minus15 10minus10 10minus5 100

120576

h = 164 n = 170100

10minus10

10minus20

Max

erro

r

(a)

10minus15 10minus10 10minus5 100

120576

h = 1128 n = 200100

10minus10

10minus20

Max

erro

r

||120577h||(G119873119878)||120577h||(G119878)

(b)

Figure 4 Dependence on 120576 for ℎminus1 = 64 128

6 Numerical Results

Let 119866 be L-shaped and defined as follows

119866 = (119909 119910) minus1 lt 119909 lt 1 minus1 lt 119910 lt 1 Ω (47)

where Ω = (119909 119910) 0 le 119909 le 1 minus1 le 119910 le 0 and 120574 is theboundary of 119866

We consider the following problem

Δ119906 = 0 in 119866

119906 = V (119903 120579) on 120574

(48)

where

V (119903 120579) = 11990323 sin(

2

3120579) + 00051119903

163 cos(16

3120579) (49)

is the exact solution of this problemWe choose a ldquosingularrdquo part of 119866 as

119866119878= (119909 119910) minus05 lt 119909 lt 05 minus05 lt 119910 lt 05 Ω

1 (50)

where Ω1= (119909 119910) 0 le 119909 le 05 minus05 le 119910 le 0 Then

119866119873119878

= 119866 119866119878 is a ldquononsingularrdquo part of 119866

The given domain 119866 is covered by four overlappingrectangles Π

119896 119896 = 1 4 and by the block sector 119879

3

1

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

0 50 100 150 200 250n

10minus8

10minus7

10minus6

Max

erro

r

h = 116

(a)

0 50 100 150 200 250n

Max

erro

r

10minus10

10minus8

10minus6h = 132

(b)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 164

||120577h||(G119873119878)||120577h||(G119878)

(c)

0 50 100 150 200 250n

10minus15

10minus10

10minus5

Max

erro

r

h = 1128

||120577h||(G119873119878)||120577h||(G119878)

(d)

Figure 5 Maximum error depending on the number of quadrature nodes 119899

0

02

04

06

08

U120576 h

0 0

0505

minus05 minus05x-axis

y-axis

(a)

0

02

04

06

08

u

0 0

0505

minus05 minus05x-axis

y-axis

(b)

Figure 6 The approximate solution 119880120576

ℎand the exact solution 119906 in the ldquosingularrdquo part for 120576 = 5 times 10

minus13

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

0

2

4

6

8times10minus11

0 0

0505

minus05 minus05

|120577h| (G119878)

x-axisy-axis

Figure 7 The error function in ldquosingularrdquo part when 120576 = 5 times 10minus13

minus2

minus15

minus1

minus05

0

120597U120576 h120597x

0 0

0505

minus05 minus05 x-axisy-axis

Figure 8 120597119880120576

ℎ120597

119909in the ldquosingularrdquo part

(see Figure 2) For the boundary of 119866119878 on 119866 is the polygonalline 119905

1= 119886119887119888119889119890 The radius 119903

12of sector 1198792

1is taken as 093

According to (49) the function 119876(119903 120579) in (4) is

119876 (119903 120579) =00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

+00051

120587int

1

0

1199101199058119889119905

(119905 minus 119909)2+ 1199102

(51)

where 119909 = 11990323 cos(21205793) and 119910 = 119903

23 sin(21205793) Since wehave only one singular point we omit subindices in (51) Wecalculate the values 119876

120576(119903

12 120579

119902) and 119876

120576(119903 120579) on the grids 119905

ℎ

1

with an accuracy of 120576using the quadrature formulae proposedin [10]

On the basis of (46) and (51) for the stress intensity factorwe have

120591120576

1119899=

00102

7120587+

2

119899(093)23

119899

sum

119902=1

(119906120576

ℎ(093 120579

119902

119895) minus 119876

119902120576

1198952) sin 2

3120579119902

119895

(52)

Taking the zero approximation 119906120576(0)

ℎ= 0 the results of

realization of the Schwarz iteration (see [2]) for the solutionof the problem (48) are given in Tables 1 2 3 and 4 Tables

0

05

1

15

2

120597U120576 h120597y

0 0

0505

minus05 minus05 x-axisy-axis

Figure 9 120597119880120576

ℎ120597

119910in the ldquosingularrdquo part

40 60 80 100 120 140 160 180 200 2200

5

10

15

20

25

30

n

n = 80n = 90n = 100n = 130n = 150

n = 170n = 180n = 200n = 220

Rϱ G119878

Figure 10 R984858

119866119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

0

5

10

15

20

25

40 60 80 100 120 140 160 180 200 220n

n = 80

n = 100

n = 120

n = 130

n = 140

n = 170

n = 190

n = 200

n = 220

Rϱ G119873119878

Figure 11R984858

119866119873119878 when 984858 = 5 by fixing 119899 for ℎminus1 = 32 for different 119899

values of ℎminus1 = 64

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

1 and 2 represent the order of convergence Table 6 shows ahighly accurate approximation of the stress intensity factorby the proposed fourth order BGM

R984858

119866119873119878 =

max119866119873119878

1003816100381610038161003816119906120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119873119878

100381610038161003816100381610038161199061205762minus(984858+1) minus 119906

10038161003816100381610038161003816

(53)

in the ldquononsingularrdquo and the order of convergence

R984858

119866119878 =

max1198661198781003816100381610038161003816119880

120576

2minus984858 minus 119906

1003816100381610038161003816

max119866119878

10038161003816100381610038161003816119880120576

2minus(984858+1) minus 119906

10038161003816100381610038161003816

(54)

in the ldquosingularrdquo parts of 119866 respectively for 120576 = 5 times

10minus13 where 984858 is a positive integer In Table 3 the minimal

values over the pairs (ℎminus1 119899) of the errors in maximum

norm of the approximate solution when 120576 = 5 times 10minus13

are presented The similar values of errors for the first-orderderivatives are presented in Table 4 when 120597119876120597119909 and 120597119876120597119910

are approximated by fourth-order central difference formulaon 119866

119878 for 119903 ge 02 For 119903 lt 02 the order of errors decreasesdown to 10

minus6 which are presented in Table 5 This happensbecause the integrands in (51) are not sufficiently smooth forfourth-order differentiation formulaTheorder of accuracy ofthe derivatives for 119903 lt 02 can be increased if we use similarquadrature rules which we used for the integrals in (51) forthe derivatives of integrands also

Figures 3 and 4 show the dependence on 120576 for differentmesh steps ℎ Figure 5 demonstrates the convergence of theBGM with respect to the number of quadrature nodes fordifferent mesh steps ℎ The approximate solution and theexact solution in the ldquosingularrdquo part are given in Figure 6 toillustrate the accuracy of the BGMTheerror of the block-gridsolution when the function 119876(119903 120579) in (51) is calculated withan accuracy of 120576 = 5 times 10

minus13 is presented in Figure 7 Figures8 and 9 show the singular behaviour of the first-order partialderivatives in the ldquosingularrdquo part The ratios R984858

119866119878 and R

984858

119866119873119878

when 984858 = 5with respect to different 119899 values for ℎminus1 = 64 andfor a fixed value of 119899 of ℎminus1 = 32 are illustrated in Figures10 and 11 respectively These ratios show that the order ofthe convergence in both the ldquosingularrdquo and the ldquononsingularrdquoparts is asymptotically equal to 16 when 119899 is kept fixed forℎminus1

= 32 and it is selected as large as possible (119899 gt 100) forℎminus1

= 64

7 Conclusions

In the block-grid method (BGM) for solving Laplacersquosequation the restriction on the boundary functions to bealgebraic polynomials on the sides of the polygon causing thesingular vertices is removed This condition is replaced bythe functions from the Holder classes 119862

4120582 0 lt 120582 lt 1 Inthe integral representations around singular vertices (on theldquosingularrdquo part) which are combined with the 9-point finitedifference equations on the ldquononsingularrdquo part of the polygonthe boundary conditions are taken into account with the helpof integrals of Poisson type for a half-plane To connect thesubsystems a homogeneous fourth-order gluing operator isused It is proved that the final uniform error is of order

119874(ℎ4+ 120576) where 120576 is the error of the approximation of

the mentioned integrals and ℎ is the mesh step For the 119901-order derivatives (119901 = 0 1 ) of the difference between theapproximate and the exact solutions in each ldquosingularrdquo part119874((ℎ

4+120576)119903

1120572119895minus119901

119895) order is obtainedThemethod is illustrated

in solving the problem in L-shaped polygon with the cornersingularity Dependence of the approximate solution and itserrors on 120576 ℎ and the number of quadrature nodes 119899 aredemonstrated Furthermore by the constructed approximatesolution on the ldquosingularrdquo part of the polygon a highlyaccurate formula for the stress intensity factor is given

From the error estimation formula (33) of Theorem 7 itfollows that the error of the approximate solution on the blocksectors decreases as 1199031120572119895

119895(ℎ

4+ 120576) which gives an additional

accuracy of the BGM near the singular pointsThemethod and results of this paper are valid formultiply

connected polygons

References

[1] Z C Li Combined Methods for Elliptic Problems with Singu-larities Interfaces and Infinities Kluwer Academic PublishersDordrecht The Netherlands 1998

[2] AADosiyev ldquoThehigh accurate block-gridmethod for solvingLaplacersquos boundary value problem with singularitiesrdquo SIAMJournal on Numerical Analysis vol 42 no 1 pp 153ndash178 2004

[3] A A Dosiyev ldquoA block-grid method for increasing accuracyin the solution of the Laplace equa-tion on polygonsrdquo RussianAcademy of Sciences vol 45 no 2 pp 396ndash399 1992

[4] A A Dosiyev ldquoA block-grid method of increased accuracy forsolving Dirichletrsquos problem forLaplacersquos equation on polygonsrdquoComputational Mathematics and Mathematical Physics vol 34no 5 pp 591ndash604 1994

[5] A A Dosiyev and S Cival ldquoA difference-analytical method forsolving Laplacersquos boundary valueproblem with singularitiesrdquo inProceedings of Conference Dynamical Systems and Applicationspp 339ndash360 Antalya Turkey July 2004

[6] A A Dosiyev and S Cival ldquoA combined method for solvingLaplacersquos boundary value problem with singularitiesrdquo Interna-tional Journal of Pure and Applied Mathematics vol 21 no 3pp 353ndash367 2005

[7] A A Dosiyev and S C Buranay ldquoA fourth order accuratedifference-analytical method for solving Laplacersquos boundaryvalue problem with singularitiesrdquo in Mathematical Methods inEngineers K Tas J A T Machado and D Baleanu Eds pp167ndash176 Springer 2007

[8] A A Dosiyev S Cival Buranay and D Subasi ldquoThe block-grid method for solving Laplacersquos equation on polygons withnonanalytic boundary conditionsrdquo Boundary Value ProblemsArticle ID 468594 22 pages 2010

[9] AADosiyev S C Buranay andD Subasi ldquoThehighly accurateblock-gridmethod in solving Laplacersquos equation for nonanalyticboundary condition with corner singularityrdquo Computers ampMathematics with Applications vol 64 no 4 pp 616ndash632 2012

[10] E A Volkov ldquoApproximate solution of Laplacersquos equation bythe block method on polygons undernonanalytic boundaryconditionsrdquo Proceedings of the Steklov Institute of Mathematicsno 4 pp 65ndash90 1993

[11] E A Volkov ldquoDifferentiability properties of solutions of bound-ary value problems for the Laplaceand Poisson equations on a

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

rectanglerdquo Proceedings of the Steklov Institute of Mathematicsvol 77 pp 101ndash126 1965

[12] A A Samarskii The Theory of Difference Schemes vol 240 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 2001

[13] A A Dosiyev ldquoOn the maximum error in the solution ofLaplace equation by finite difference methodrdquo InternationalJournal of Pure and Applied Mathematics vol 7 no 2 pp 229ndash241 2003

[14] A A Dosiyev ldquoA fourth order accurate composite gridsmethodfor solving Laplacersquos boundary value problems with singular-itiesrdquo Computational Mathematics and Mathematical Physicsvol 42 no 6 pp 867ndash884 2002

[15] E A Volkov ldquoAn exponentially converging method for solvingLaplacersquos equation on polygonsrdquo Mathematics of the USSR-Sbornik vol 37 no 3 pp 295ndash325 1980

[16] E A Volkov Block Method for Solving the Laplace Equation andfor Constructing Conformal Mappings CRC Press Boca RatonFla USA 1994

[17] G J Fix S Gulati and G I Wakoff ldquoOn the use of singularfunctions with finite element approximationsrdquo Journal of Com-putational Physics vol 13 pp 209ndash228 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of