Jacobi and Gauss Seidel Methods To Solve Elliptic Partial ... and... · Volume 15 دذعلا October 2018 ربوتكا International Science and Technology Journal ةينقتلاو

Volume 15 العذد

2018October اكتوبر

International

Science and Technology Journal

المجلة الدولية للعلوم والتقنية

حقوق الطبع محفوظة للمجلة الدولية للعلوم والتقنية

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Jacobi and Gauss Seidel Methods To Solve Elliptic

Partial Differential Equations

Mohamed Mohamed Elgezzon

Higher Institute for Engineering Technology-- Zliten

[email protected]

Abstract

Solving elliptic partial differential equations manually takes

long time. Computer plays an important role to solve this kind

of equations. Finite element method is used to solve elliptic

partial differential equations by reducing these equations to

linear system Ax = b using iterative methods as Jacobi and

Gauss Seidel methods.

These methods are applied for two examples in this paper.

Keywords: iterative methods – elliptic partial differential

equation – finite element method.

الملخص:

انحاسىب حيث اٌ ، حم انًعادالت انتفاضهية انجزئية اننالصة يذوياً تأخذ ولتاً طىيلً

)انكًثيىتز( يهعة دوراً يهًاً نحم هذا اننىع يٍ انًعادالت وتستخذو طزيمة انعنصز

انًحذد نحم انًعادالت انتفاضهية انجزئية اننالصة تىاسطة تخفيض هذه انًعادالت إنى

تىاسطة استخذاو طزق انتكزار يثم طزيمة جاكىتي Ax = bطزق اننظاو انخطي

mailto:[email protected]



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ىرلة انثحثية يتى تطثيك انطزيمتيٍ يٍ خلل يثانيٍ وطزيمة جاوس . و في هذه ان

نذراستهًا في هذا انثحث.

Introduction.

Let be a bounded domain in Rd, with boundary . It is

assumed that f and g are continuous on . So a unique solution

exists

- = f(x,y) For(x,y) (1)

u(x,y) = g(x,y) in

= {(x,y) : a < x < b,c < y < d}.

The concept of the finite element is to apply this to equation

(1).

First, the test function v is picked where v satisfies the

boundary condition v = 0 on and multiply the first

equation by v. Finally, we integrate the equation over , using

Green's formula as

∫

= - ∫

v ds + ∫

.

A finite element discretization of (1) is based on the weak

formulation:[2] seek u V = [H

( )]

d such that.

A(u, v) = (f,v) V (2)

Where a(u, v) = ∫

, (f,v) = ∫

The approximate solution uh Vh V satisfies



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a(uh, v) = (f,v) Vh (3)

we define uh = ∑ ui i (x), thus, we substitute uh in the

following equations.[3]

a(uh, k) = (f, k) k = 1, …, N

We obtain

∑ ui .a( I, k) = (f, k) k = 1, …, N

(*)

We could write the formula in (*) as:

Ax = b. (4)

Where A Rn×n

is symmetric matrix, b Rn , and x R

n

is the request.

To solve (4) we use the iterative technique as follows

solution[4]

Xk+1

= Txk + C, k= 0,1, ….., (5)

Where x0 is given

For the convergence it is required that for all x0 R

n the

vector sequence produced by the iteration (5) should be

convergent to the same vector x* R

n. And x

* would be

approximate solution of x in (4), that is, Ax* = b.

The following two theorems are needed for the seek of

completeness.[5]



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Theorem 1. Assume that (M,N) is a splitting of the regular

matrix A, such that T = M-1

N is convergent matrix. Then the

iterative method (5) based on this splitting is convergent to the

solution of the system of linear algebraic equations Ax = b.

Theorem 2. Assume that A is monotone matrix, Then any

regular splitting of A generates a convergent iterative method,

that is, if (M,N) is a regular splitting of A then p(M-1

N) < 1.

In the following, some well-known iterative methods are

considered to the solution of (4) such as, Jacobi method and

Gauss Seidel method.

(1) Jacobi iterative method (JIM).

The Jacobi iteration of the form:

X

=

bi - ∑

aij X

i = 1,2,……,n,

k= 0,1,2,….. (6)

Is considered, where it is assumed that aii 0. Introducing the

notations[6]

D = diag A;

-U = upper triangular part of A;

-L = lower triangular part of A;

We have the obvious relation

A = D – U – L.



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Then the Jacobi iterative method (6) can be rewritten in the

matrix form

Xk+1

= D-1

(b + (U + L) xk), (7 )

That is,

Xk+1

= D-1

(U+L)xk + D

-bb.

Obviously, this is a BIM with GJ = D-1

(U+L) and cJ = D-1

b.

On the other hand,

Dxk+1

= (U+L) XK + b.

That is, if MJ = D and NJ = U+L is defined then

MJ xk+1

= NJ xk + b.

Moreover, the relation MJ – NJ = D – U – L = A holds.

Consequently, the iterative method of the form (7) is BIM

based on the splitting

A = MJ – NJ.

(2) Gauss Seidel iterative method (GSIM).

The Gauss Seidel iteration of the form

Xik+1

=

bi - ∑

aij x Xj

k+1 - ∑ aij x Xj

k i=

1,2,….n, k= 0,1,2 …. (8)

The Gauss Seidel method can be rewritten in the matrix form

Dxk+1

= Lxk+1

+ Uxk + b.



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It can be remarked that the main difference between the Gauss-

Seidel method and the Jacobi method is the following:[7][8]

Using the Jacobi method for the computation of the new

iteration the previous approximations are applied in each

equation, while in Gauss Seidel the new already computer in

more suitable form is rewritten as

(D - L)xk+1

= Uxk + b. (9)

Introducing the notations

MGS = D – L, NGS = U,

We obtain

MGS – NGS = D – L – U = A.

That is, MGS – NGS is a splitting of A. Then (9) can be rewritten

in the form

MGSXK+1

= NGSXK + b; or

Xk+1

= TGSXK + CGS;

Where, we have used the notations

TGS = MGS-1

NGS = (D-L)-1

U, CGS = MGS-1

b = (D-L)-1

b.

Example 1

Consider the follows problem with boundary conditions and

given domain

- = 2 – (x2 + y

2) 0 x,y 1



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u(x,y) = 0 when x= 1, y = 1;

u(x,y) =

(1-y

2) when x=0;

u(x,y) =

(1-x

2) when y=0;

Example 2

Consider the follows problem with boundary conditions and

given domain

- = 2 – (x2 + y

2) 0 x,y 1

u(x,y) = 0 in boundary

Briefly the solution of the given problem by theoretical

methods is applied as defining step size h and partitioning the

intervals [a,b] and [c,d] into n equal parts and the finite

element method.[9]



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Figure 1. The shape of example 1 where n=25

Figure 2. The final shape of matrix A of example 1 where n= 10



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For example 1 Ax = b when n = 2

A=

[ ]

, b =

[

]

x =

[

]



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Figure 3. The shape of example 2 where = 25

Figure.4. The final shape of matrix A of example 2 where n = 20



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For example 2 Ax = b when n = 2

=

[ ]

, b =

[

]

x =

[

]

The results of numerical tests of the Jacobi an gauss seidel

iterative methods for example 1 and example 2 are given in the

following tables:

Table 1 Iteration and residual of matrix A by Jacobi and Gauss

Seidel method of example 1

20/1 10/1 5/1 h

Residual Iteration Residual Iteration Residual Iteration

9.931e-4 525 9.875e-4 131 9.592e-4 32 JIM

9.779e-4 288 9.590e-4 51 6.611e-4 18 GSIM



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Table 2 Iteration and residual of matrix A by Jacobi and Gauss

Seidel method of example 2

20/1 10/1 5/1 h

Residual Iteration Residual Iteration Residual Iteration

9.931e-4 525 9.868e-4 109 9.965e-4 22 JIM

9.779e-4 288 9.670e-4 22 7.071e-4 12 GSIM

References

[1]. Aleyan O. A.: On matrix splitting and application into

iterative methods for linear systems An Academic. Intellectual,

Cultural, Comprehensive and Arbitrated Journal Issued by the

Faculty of Arts and Sciences in Zliten, Al- Merqab. Issue No.

18 2009 (JUNE) 1377.

[2]. Berman, A. and plemmons, R. Nonnegative matrices in

the mathematical sciences, Academic press, New York, 1979.

[3]. Hom, R.A. and Johnson, C.R. Matrix analysis, Cambridge

University press, Cambridge, 1986.

[4]. Ortega, J.M. and Rheinboldt, W.C. Iterative solution of

linear systems, plenum press, New York, 1988.

[5]. Miroslav Fiedler., Special matrices and their applications

in numerical mathematics, Kluwer, 1986.

[6]. Miroslav Fiedler and Vlastimil ptak, On matrices with

non- positive off- diagonal elements and positive principal

minors, Czechosl. Math. Journal (87) 12, 1962, pp 382-400.



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[7]. Varga, R.S. Matrix iterative analysis, prentice Hall, New

York, 1962.

[8]. Young, D. Iterative solution of large linear systems,

Academic press, New York, 1977.

[9]. Richard L. Burden and J. Donglas Faires Numerical

analysis, third edition prindle, weber & Schmidt, Boston, 1985.

Documents

Jacobi and Gauss Seidel Methods To Solve Elliptic Partial ... and... · Volume 15 دذعلا October 2018 ربوتكا International Science and Technology Journal ةينقتلاو