COMMUNICATIONS IN APPLIED NUMERICAL METHODS, VOl. 8, 161-169 (1992)

ON THE CONVERGENCE RATE OF AN ADAPTIVE GRID PATTERN FOR RESOLVING POINT SINGULARITIES IN

ELLIPTIC PROBLEMS

P. R. DEVLOO INPE/DEM - Instituto de Pesquisas Espaciais, Av.Dos Astronautas, 1758 - Caixa Postal 515-12201 Sao Jose,

Dos Campos, SP Brazil

INTRODUCTION

This article analyses an inexpensive solution process for a certain class of elliptic partial differential equations with a singular solution. The modelling of singular solutions is a very frequent occurrence in the engineering modelling world. As the most common problems we can cite: the modelling of heat fluxes in domains with re-entrant corners, the modelling of inert masses attached at discrete points on plates, the modelling of stress intensity factors for crack analysis, etc. In most cases, the engineering approach to these models is to disregard the solutions around the singularity, but expect the finite-element approximation to yield acceptable results away from the singularity. If the mesh generator permits, a denser finite- element grid is generated in the vicinity of the singularity.

In this work we analyse a predefined adaptive grid pattern to better approximate the singularity in the domain. It is shown that the adaptive grid pattern not only improves the accuracy of the approximation in the vicinity of the singularity, but also the accuracy in domains away from the singularity. The technique presented also has the advantage of being easy to incorporate in existing finite-element codes.

PROBLEM STATEMENT

Assume we seek a finite-element approximation of the problem shown in Figure 1, where J(s) is chosen such that

0 u = Jr sin - 2

is the solution of the model problem. This kind of singularity can be recognized in many physical problems including electrical field problems, slit membrane problems, etc. The results described for this model problem also easily extend to the elasticity problem involving crack tips.

For a more complete account of singular solutions to elliptic problems, the reader is referred to References 2 and 3.

0748-8025/92/030 161 -09$05 .OO 0 1992 by John Wiley & Sons, Ltd.

Received 23 May 1990 Revised 3 May 1991

162 P. R. DEVLOO

du - = F(S) dn

Figure 1. Model problem

It can easily be verified that

V u = --sin - lx+-cos (!)b 2 k (3 2,/r

and that

Clearly, the solution u has a square root singularity at the origin, which implies that the finite-element approximation using uniform meshes will converge at a suboptimal rate in the vicinity of the crack tip:

(4)

In order to recover a reasonably accurate finite-element approximation, the analyst can use

1. self-adaptive finite-element techniques 2 . graded meshes in the vicinity of the singularity.

Whereas the first approach has the definite advantage of being totally automated, it is disappointing to know that the adaptive algorithm will perform the obvious at considerable computational expense: it will estimate the error and detect that the error is largest in the vicinity of the singularity. It will then refine the grid around the singularity.

The second approach has the disadvantage that it is cumbersome to generate graded grids and that there is no indication as to the required accuracy of the initial approximation.

In this work we will analyse the convergence rate of a predefined adaptive grid pattern. Such a pattern is easy to generate and improves the accuracy of the solution by a significant factor.

( 1 24 - uk ( (H' (P) < C,/h I u (H3/'

the following tools:

CONVERGENCE

When observing a grid which has adaptively been refined around a singularity, one recognizes a regular repeated pattern in the grid structure (see Figure 2) . The underlying idea of the convergence proof is to take advantage of this repeating pattern in the grid structure. The adaptive grid will be decomposed in different layers and the total error of the adaptive grid will then be taken as the sum of the error of the individual layers.

The basic interpolation error estimate for finite-element approximations is

( 1 u - uh I ) H ' ( O ) < c h (H*(P) ( 5 )

CONVERGENCE RATE OF ADAPTIVE GRID PATTERN 163

Figure 2. Adaptive grid with regular pattern

Now N

I = 1 (6) 2 1 ) 2.4 - uh Ilk(0) = 11 u - uh 1\&'(0,) + 11 - uh IlH'(i2,)

where QI is the area of layer I and Qi is the innermost patch of elements. For each patch we can write

1 1 u - u h 11&' ( * r , < c h 2 1 1 u Ilk(*,)

I u I&(*,) < I u ILW)

(7)

The layer QI is included in a semicircle segment with radius h < r < 2d2h. Therefore

(8)

where Q[ is the domain corresponding to the semicircle h c r c 2d2h (see Figure 3). The H 2 seminorm in the larger domain QI' is equal to

2 J 2 h

h I u I&(*;) = 1, ( u z + 2uL + u:y)r dr d8 (9)

but as

2 2 1 2 uxx = cos Burr - - sin 8 cos 8u,e + sin 8uee r r

and similar expressions can be found for u, and uyy, it is easily verified that

Figure 3. Finite-element domain against interpolation domain

164 P. R. DEVLOO

Based on the exact solution (see equation 1 ) we find:

1 0 urr = - - (r)-3'2 sin - 4 2

1 0 Ure = - - (r)-3'2 cos -

4 2

sin - 4 2 1 -3/2 0 uee = - - (r)

such that

1 < C - hi

Therefore, for each layer of elements,

h i hi

11 U - uh Ilh'(~() < c - = Chr

where hr is the characteristic length of an element within the element layer. Denote by h the mesh size of the outermost layer of elements:

h = ho

then

where CY is the mesh size reduction factor between two subsequent layers of elements. The sum of the interpolation errors of the different layers is equal to

N

1 1 u - uh < c C hl l = O

CONVERGENCE RATE OF ADAPTIVE GRID PATTERN 165

In our example a = 2, and therefore the total error of the layers of elements will be smaller than or equal to two times the interpolation error of the outermost layer.

For the innermost elements, the interpolated function is not sufficiently regular to use the interpolation error estimate used so far, but for the type of singularity considered, we have O ( / h ) convergence:

Therefore the sum of the interpolation error for all layers and the inner elements is equal to

where CI and CZ are independent of the number of layers but are not necessarily equal. Depending on the relative magnitudes of the constants, for a definite number of layers the interpolation error will be dominated by the error of the inner elements or the error in the layer of elements.

It is important to note that, based on this derivation, we can conclude that, if the number of layers is found to be sufficient for one particular mesh size, then this number of layers will be sufficient for any mesh size.

NUMERICAL RESULTS

Several numerical tests were made with the model problem described above (see equation (1)). The initial grid consists of two elements. This initial grid is then uniformly refined a variable number of times. For every grid, the elements around the singularity are replaced by the recursive element consisting of 0 ,1 ,2 ,4 ,8 ,16 and 32 layers of elements. We then study various errors (true errors) for the different configurations.

Figure 4 shows the total energy error for the different configurations. The vertical axis corresponds to the error on a logarithmic scale. The horizontal axis corresponds to the number of uniform refinements applied to the initial grid. From this figure we conclude that:

1. the adaptive grid pattern does not improve the convergence rate of the finite-element

2. the error decreases as the number of layers increases, 3. the error only decreases up to an adaptive grid pattern with eight layers. Beyond eight

layers, the error does not decrease significantly, 4. the error decreases by a constant factor when using the adaptive grid pattern. At virtually

no additional cost, the total error of the finite element is decreased by approximately 75 per cent, irrespective of the size of the external grid.

approximation. The rate of convergence is still O(dh),

Figure 5 shows the sum of the error in the layers and the error of the innermost elements. O ( / h ) rate of convergence is observed as predicted by the theory. By increasing the number of layers up to eight, the error is decreased. For more than eight layers, the decrease in error is unnoticeable.

Figure 6 shows the evolution of the error in a fixed domain which does not contain the singularity. It is noted that the optimal rate of convergence O(h) is recovered for fixed domains not containing the singularity. Again, the use of the adaptive grid pattern decreases the error by a constant factor, which is smaller than the factor observed for the total error.

166 P. R. DEVLOO

E r r o r

no layer 1 layer 2 layers

4 layera 8 layers 16 layers 32 layers

0 1 2 3 4 6

Number of global Isvele of rrflnement

Figure 4. Total energy error

Figure 7 shows the ratio of the error in the layers of elements around the singularity over the error of the domain external to the layers. Increasing the number of layers up to eight decreases the ratio by a constant factor. As the number of global refinements is increased, the ratio tends to a constant, irrespective of the number of layers. For the adaptive grid pattern with eight layers, the error in the layers accounts for 50 per cent of the total discretization

1

- nolayer --t- I layer - Zlayen - )layers - 8 layerr --9- 16 layerr -t- 321ayen

.1 E r r o r

01 I * , ‘ 1 ’ I ’ 1 *

0 1 2 3 4 5 6

Number of global levels of reflnement

Figure 5. Error in the layers

CONVERGENCERATEOFADAPTIVEGRIDPATTERN 167

-Q- nolayer - 1 layer - 2layers - 4layers - 8 layers I 16layers - 32layers

1 2 3 4 5 8 .OOl 1 . I . I ' 1 . 7

0

Number ot global rsl lnsmentr

Figure 6. Error in the external level 1 domain 4

error. For the approximation without layers, the error of the elements around the singularity accounts for 75 per cent of the discretization error.

Based on the observed global rate of convergence, and the convergence rate estimate for the layer, the ratio of the error in the layer over the error in the external elements can be predicted in the following manner.

Err L r y d E r r - . .

1 o ! . I . 1 . l ' l . 1 '

0 1 2 3 4 5

* nolayer - 1 layer - 2 layen

7 4 layers _I 8layers --O- 16 layers - 32 layers

Number ot Qlobd Reflnementr

Figure 7. Ratio of the error in the layers over the error in the external domain

168 P. R. DEVLOO

addltlon to external domaln after glooal reflnernent external dornaln before global reflnement

[ZI Constralned nodes before global reflnement

element layers after global refinement

Figure 8. Different domains after a global refinement

From the following definitions:

0 en = total energy error for i global refinements 0 err = energy error in the layers for i global refinements 0 eE, = energy error in the domain external to the layers for i global refinements

2 e?, = eti + e E l

Assume

then, from Figure 8 we see that the reduction of the total error for one uniform refinement level to the next is computed by adding three distinct contributions;

1. The error of the elements of the external domain at level i is reduced by a factor of 4

2. The error of the elements which have been added to the external domain of level i + 1

3. The error of the layers at level i + 1 is equal to one-half the error of the layers at level i:

(the optimal convergence rate).

is equal to one-half the error of the layers at level i.

er; 2 2 2 eL; eLi eE; = - = - + - + - e!c;+ 2 2 2 4

therefore 2 2 2 1 e L ; eL; e E ; - (eri+ eE;l =-+-+-

2 2 2 4

(14)

The difference between the predicted ratio and the computed ratio can be explained by the loss of accuracy caused by the constrained nodes.

ACKNOWLEDGEMENT

The support of the author from CNPq under grant number 400583/89-5 is gratefully acknowledged.

CONVERGENCE RATE OF ADAPTIVE GRID PATTERN 169

REFERENCES 1. L. Demkowicz, P. Devloo and J. T. Oden, ‘On an h-type mesh refinement strategy based on a

minimization of interpolation error’, Comput. Methods Appf. Mech. Eng., 53(3), 67-89 (1983). 2. I. Babuska and A. Miller, ‘The post-processing approach in the finite element method - Part 2: the

calculation of stress intensity factors’, Int. j. numer. methods eng., 20, 11 11-1 129 (1984). 3. P. Grisvard, ‘Boundary value problems in non-smooth domains’, Lecture Notes No. 19, Department

of Mathematics, University of Maryland, 1980. 4. P. Devloo, ‘Recursive elements, an inexpensive solution process for resolving point singularities in

elliptic problems’, TICOM Report 89-10, The University of Texas at Austin, September 1989.