Adaptive multigrid method for numericalsolutions of elastic wave equation

Bo Han *, Xiaoyan Zhou, Jiaqi Liu

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Abstract

Finite-element multigrid method is applied to two-dimensional elastic wave equation

in frequency domain. To avoid nonsensical increase in computational cost caused by

global grid refinement, an adaptive algorithm is designed. Finally, numerical tests are

carried out.

� 2002 Elsevier Science Inc. All rights reserved.

Keywords: Elastic wave equation; Multigrid method; Adaptive algorithm

1. Introduction

Efficiently solving the numerical solution of wave equation is the basis ofwave-field simulation and inversion. Ordinary methods, such as differencemethod [5], finite-element method [7], and spectrum method [6], often failed orare inefficient when the solutions have oddness or oscillation. Multigridmethod (MGM), developed in the early 1970s, is a kind of fast method forsolving boundary value problem. It has been widely applied to many fields suchas integral equation, eigenvalue problem, ordinary nonlinear equation, and soon (see [1,4]). [2] successfully used it as the forward method, thereby greatlyimproved the efficiency of the GPST inversion algorithm. Since the domain,simulated by elastic wave equation, is complicated in practice, the distributionof parameters is irregular or the solution has oddness. Therefore differentdiscrete steps are needed to deal with the problem, and it is necessary to use the

* Corresponding author.

0096-3003/02/$ - see front matter � 2002 Elsevier Science Inc. All rights reserved.

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Applied Mathematics and Computation 133 (2002) 609–614www.elsevier.com/locate/amc

strategy of adaptive grid refinement [3,8]. Namely the grid is locally refined bycomparing a given error with the computational error. It is a dynamic process.

In this paper, an adaptive MGM is designed for two-dimensional elasticwave equation in frequency domain. The rest of the paper is organized asfollows. In Section 2 we discuss the basic finite-element method. The MGMis described in Section 3. In Section 4, an adaptive algorithm is introduced.Finally, in Section 5 numerical tests are carried out.

2. Finite-element method

Consider the following two-dimensional elastic wave equation in frequencydomain

o

oxkðx; zÞ oU

ox

� �þ o

ozkðx; zÞ oU

oz

� �� s2qU ¼ F ðx; z; sÞ;

ðx; zÞ 2 X; sP 0;

Uðx; z; sÞjx¼�L ¼ U1ðz; sÞ; Uðx; z; sÞjx¼L ¼ U2ðz; sÞ;o

onUðx; z; sÞjz¼0 ¼ 0; Uðx; z; sÞjz¼H ¼ U3ðx; zÞ;

ð1Þ

where X ¼ ½�L; L ½0;H ; z ¼ 0 is the earth’s surface; F ðx; z; sÞ is the sourcefunction; qðx; zÞ is the density function; kðx; zÞ is the elastic parameter function;/1;/2, and /3 are far field conditions, approximately zero.

For any fixed frequency s > 0, according to variational principle, the vari-ational problem, equivalent to (1), is

I ½U ¼ � 1

2

ZX

koUox

� �2"(

þ oUoz

� �2#þ s2qU 2

)dxdz

�Z

XUF dxdz: ð2Þ

The domain X is dissected by a rectangle in such a way that: if h is the stepsize of a finer grid (for simplicity we omit the subscript in x and z directions),then 2h is the step size of the subfiner grid. X0 is the coarsest initial dissection.Then we obtain a sequence of dissections Xi, i ¼ 0;M . For every Xi there existsa finite-element subspace HðXiÞ, satisfying the inclusion relation

HðX0Þ � HðX1Þ � � � � � HðXMÞ ð3Þ

The variational problem (2) is to be solved in HðXMÞ. Suppose the finestlayer XM has NM rectangular single elements. In canonical single element e, forany fixed frequency s. Let

610 B. Han et al. / Appl. Math. Comput. 133 (2002) 609–614

Uðx; z; sÞ ¼ aþ bxþ cz;

kðx; zÞ ¼ keðcons tan tÞ;

qðx; zÞ ¼ qeðcons tan tÞ:

Then carrying out the ordinary stiffness analysis and population matrix com-position, we obtain

KMUM þ s2 �KKMUM ¼ �FFM ; ð4Þ

where KM and �KKM are the stiffness matrix and mass matrix, respectively. Theyare symmetric, positive definite and sparse. KM and �KKM at most have seven non-zero diagonal lines. Therefore (4) can be quickly solved by the MGM.

3. Multigrid method

For simplicity, the linear equation (4) is rewritten as

LMUM ¼ FM in XM : ð5Þ

Under coarser subdivisions, the equations analogous to (5) are denoted by

LmUm ¼ Fm in Xm; m ¼ M � 1; . . . ; 1; 0: ð6Þ

The distinguishing feature of MGM lies in that local treatment on differentscales is connected with the interrelation between every scale by using theadministrative structure of finite-element subdivision. The restricted operatorrm and the prolongated operator pm of the MGM are defined by

rm: UðXmÞ ! UðXm�1Þ; pm: UðXm�1Þ ! UðQmÞ; ð7Þ

where UðXmÞ and UðXm�1Þ are the grid spaces corresponding to the dissectionsXm and Xm�1, respectively. Corresponding to rectangle dissection, rm and pm arethe nine-point restriction operator and prolongation operators [4], respectively.

The coarser grid matrix is defined by

Lm�1 ¼ rmLmpm; m ¼ M ; 1: ð8Þ

The right-hand side is given by

Fm�1 ¼ rmðFm � LmUmÞ; m ¼ M ; 1: ð9Þ

The incomplete LU decomposition is adopted as the smoothing iteration of theMGM.

B. Han et al. / Appl. Math. Comput. 133 (2002) 609–614 611

4. Adaptive algorithm

The key to the adaptive algorithm lies in how to judge the domain that needsto be refined; it is fulfilled through a posteriori error estimates to conductadaptive slow treatment.

Since the solution of (5) cannot be globally smooth, the dissection cannot begood in the whole domain. A convergence, lower than the average error, will begenerated, called ‘‘slow convergence’’.

Therefore a posteriori error estimate must be defined, so as to give a simplecriterion De in every simply element. If we use Uh to denote the approximatesolution when the step of the dissection in h, then De should be only interre-lated to the approximate solution Uh and F. The slowest convergent element e0

can be given through comparing the size of De. Thereupon the efficiency can beimproved through local treatment on the bad element e0, and only local re-finement is needed.

The posteriori error estimate is defined by

DD ¼ kp2hrhU � rhUk2D; ð10Þ

where D is an inner subdomain or only a part of the smooth boundary of X, rh

and p2h are the projection operator and interpolation operator, correspondingto the step size h and 2h, respectively.

The adaptive slow treatment is carried out through comparing the posteriorierror estimates between the mth and ðm� 1Þth layers. For the elementEið16 i6 maxÞ of the ðm� 1Þth layer, since Hm�1 � Hm. There are J elementsof Hm which are included in Eið16 J 6 4Þ, denoted by eij ðj ¼ 1; . . . ; JÞ. Let

Gi ¼ DEi;

where Ei ¼SJ

j¼1 eij, DEi is given by (9). Search out the bad subdomain, and letGi0 ¼ maxi Gi. Then a new dissection can be obtained by doubling the step sizeof the bad element and then a proper treatment, through the finite-elementdissection criterion, can be realized. The procedure is repeated until DEi 6 e,where e is the precision prior given.

5. Numerical experiments

Choose kðx; zÞ ¼ 1, qðx; zÞ ¼ 1, U1 ¼ U2 ¼ U3 ¼ 0. Two numerical tests ofthe adaptive MGM are given.

Example 1. X ¼ ½0; 3 ½0; 2, s ¼ 10, the analytic solution is

Uðx; zÞ ¼ xðx� 3Þzðz� 2Þ:

612 B. Han et al. / Appl. Math. Comput. 133 (2002) 609–614

Example 2. X ¼ ½0; 1 ½0; 1, s ¼ 1, the analytic solution is

Uðx; zÞ ¼ ðch10xþ ch10zÞ=ch10:

The adaptive procedure is presented in the following:Step 1. Give the coarsest grid and conduct the canonical dissection. Then

solve the problem by the finite-element MGM.Step 2. Judge the worst element by the posteriori error estimate, refine it by

half-step, and conduct the proper treatment through the finite-ele-ment dissection criterion to obtain a new dissection.

Step 3. Repeat the procedure, until the energy norm of every elementDEi 6 10�2.

Number of dots and CPU time of the basic MGM and the adaptive algo-rithm are tabulated in Tables 1 and 2, respectively. The grids of the two ex-amples are plotted in Figs. 1 and 2.

Table 1

Number of dots and CPU time of MGM

Example Dot number of the

coarsest grid

Dot number of the

finest grid

CPU time (s)

1 4 3 ¼ 12 49 33 ¼ 1617 5.230

2 5 5 ¼ 25 33 33 ¼ 1089 4.998

Table 2

Number of dots and CPU time of adaptive algorithm

Example Dot number of the

coarsest grid

Dot number of the

finest grid

CPU time (s)

1 4 3 ¼ 12 12 9 ¼ 108 0.858

2 5 5 ¼ 25 17 17 ¼ 289 1.998

Fig. 1. The finest grid of adaptive algorithm of Example 1.

B. Han et al. / Appl. Math. Comput. 133 (2002) 609–614 613

The numerical results indicate that for the smoother case of Example 1, auniform dissection is given by the adaptive algorithm, but for the case ofExample 2, the grid is automatically refined by the adaptive algorithm near oddpoints, and global refinement is avoided.

Acknowledgements

The work was supported by the National Nature Science Foundation ofChina under grant No. 19971026, and the Bashidian Foundation of ChineseEducation Ministry under grant No. 1999021311.

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Fig. 2. The finest grid of adaptive algorithm of Example 2.

614 B. Han et al. / Appl. Math. Comput. 133 (2002) 609–614