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PubL RIMS, Kyoto Univ.13 (1977), 47-60

Convergence of the Finite Element MethodApplied to the Eigenvalue ProblemBy

Kazuo ISHIHARA*

I. Introduction

This paper is concerned with the finite element approximationschemes for the eigenvalue problem:(1) Au+fai=Q in with the boundary condition

= 0 on S (Dirichlet type)or

du/dn Q on S (Neumann type)where A is the Laplacian, J2 is abounded domainin R2, S is the piecewisesmooth boundary of J2, and n is the exterior normal.

We put the equation (1) into the weak form:(2) a (u, v) I (u, v) for any v^Vwhere

a(?/,v) = I (uxvx H-tiyVy) dxdy ,

(u,v) = \ uvdxdy ,Jsand where V=H01() or V= ?() accordingas the boundary conditionis Dirichlet type or Neumann type. Here JT(J2) is the Sobolev spaceof order 1 and Ho1(fl) is a subset ofH1(fl) composed offunctions vanish-

Communicated by S. Hitotumatu, February 2,1976.* Department of Mathematics, Faculty of Science, Ehime University, Matsuyama790,Japan.

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48 K A Z O O I S H I HA R Aing on S. We apply to (2) the consistent mass (CM) and the lumpedmass (LM) schemes, with the linear interpolation function. Then thecorresponding equations may be written in the following forms:(3) KU M.U for the CM scheme,(4) KU=lM2U for the LM schemewhere K is the s t i f fness matrix, M1 is the consistent mass matrix andM2 is the lumped mass matrix. Strang and Fix [3] have proved thatthe approximate eigenvalues and eigenfunctions in the CM schemecon-verge with a certain rate of convergence to the exact ones. However,for the LM scheme it seems that no such proof has been publishedexplicitly.

In this paper, we propose the mixed mass (MM)scheme:(5) KU=lMUwhere

and prove the convergence of the LM and the MM schemes with acertain rateof the convergence. Finallywe give the numerical examples.The results by the three schemes show good agreements with the exactsolutions. In particular, the MM scheme demonstrates more accurate nu-merical results than the other schemes.

2. The Order of ConvergenceWe assume that every eigenvalue of (1) is distinct and simple. Fur-

ther for simplicity we assume that the domain J2 is a convex polygon.Let us divide X? into the triangles of finite numbers in the usual manner:

where m is the number of the triangular elements. The finite numberof nodes are denoted by Pl9 P2, ,Pn. We further assume that any

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F E M F O R E IG E N V A L U E P R OB L E M 49two adjacent triangles have an only common side. Now we define thefinite dimensional spaces XhC L2($) andYhdV as follows:

Xh={$:$ is a piecewise constant function in each triangularele-ment in the sense stated below},

Yh={$:$ is a function which is linear in each triangular elementand $ V\

where h is the length of the largest side of all t riangular elements.''Piecewise constant" means, for instance,

where St is defined as in Figure 1. Then there exist the basis{$l9 - ,and {0j, , $ } for Xh and Y*, respectively, such that

Every ( f > ( E X fl or i f) E Yh can be uniquely determined as=Z ?=i

where 01? , 0 n are nodal values. We say that $ Xh and $

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50 K A Z U O I S H IH A R Ah : the eigenvalues of (1) (/11

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F E M F O R E I G E N V A L U E P R O B L E M 5 1

Proof. B y t he Schwarz inequality , th e Poincare inequality andL e m m a 1 , we have

< C0 a($9$) -J~ch a($,$) + Ma ($, $)

where C0 and C m a x { C 0 \ / ~ ~ < ? - d i a m J 2 ) } are constants, which are inde-pendent of /2. This completes the proof .(*}

Lemma 4. (min-max principle) Let Sidenoteanyi-dimensionalsu bspace of V. Then, eigenvaluesa re characterized by the equation

, . a (u. ti)A * mm max - .Si u(ESi \\U\\w - f c O " "

For the CM scheme, Straiig and F ix [3] proved th e fo l lowing tw otheorems.Theorem 1. For small h and the fixed i, there exists a constant

C L -which is independent of h, such that

Theorem 2. For smallh and the fixed z,ihere exists a constantc it which is indepe?ident of h, such that

N ow ^ ve can p rove the fo l lowing results for the LM and the M Mschemes. F i rs t we give error bounds for the eigenvalues.

(*} As a matter of fact, if the boundary condition is of the Neumann type, we have toreplace a(utv) by a(utv) +r(u,v) with a positive f so that a(u,U^^T\\u\\2. How-ever, if -9 has a certain symmetry and if we consider higher modes as in thenumeri-cal examples, we can apply the Poincare inequality directly.

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FEM FOR E I G E N V A L U E P R O B L E M 53Hence we have

T U KU U KUA j= max

i +CA +Chh A *Therefore, by Theorem 1, we obtain

This completes the proof.The following two theorems give error bounds to the approximate

eigenfunctions.Theorem 4. Let { A * ,wi9 Wi} be the LM solutions. For small

h and the fixed i, there exists a constant ciy which is independentof h, such that

Ut-WiYh, a(Pu, v)=a(u, v) for any v^Y\L:V-*X\ (Lu,v) =(u,v) forany v^X\ v^Y\ v^v.

By the approximation theorem ([3]), there exists a constant Cl9 whichis independent of A, such that

Since a(Put,Wj)=a(ui9Wj) by the definition of the weak solution andsince (Lui9 W j) = (ui9 W j} , w e have

- A i)(Put, Wj)=lj (Put, wj) ~A * (Put, wi) = a(Puiy z t > j ) - Xi(Pui9W= a(ui9wi) - A /(Put,wj) =At i,wj) - (Fui9 f e > y )

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54 K A Z U O I S H IH A R A=At(Lui,w ) y )- li(Put, f e > ,) =/U(Lui-Put, w > ,)

wherePzci Pu Xh. Because the set [te) ly ,w7l]forms anorthonormalbasis for X*1, we have

Since Ai is distinct from the other eigenvalues, for sufficiently small /z,there exists a constant C2, which is independent of /?, such that

Hence, putting0=(Pui,iwi), there exists a constant C3, which is inde-pendent of h, such that

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F E M F O R E IG E N V A L U E P R O B L E M 55Theorem 5. Let {liyzi9zt} be the MM solutions. For small h

and the fixed z, there exists a constant ciyivhich isindependent of h,such that

Proof. We introduce two spaces H=LZ(&) XV andH={{u,u}:uX\uY\ v~fi}

Addition and scalar multiplication are defined in the obvious manner andthe inner product and the norm are defined by

[X v}, {w,z}']= {(u,rv) + (v9z)}9

Define two mappings P and Q as fo l lows:P:V-*Y\ a(Pu, )=a(u,) for any

for an y { z e > , w} G H .B y the approximation theorem ( [3] ) , there exists a constant Ci9 whichis independent of h, such that

Since a(Puf, Zj)=a(uiy z^) by the definition of the weak solution, andsince [Q{,,ttt}, {zJ9 %}] =[{ , Ut}, {%,%}] , we have

H i9 PUi},{ ,,,} ]

-a ,zy-t,Put}, {zj9

= A ,[{ut9u,}, {,,

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56 KAZU O ISHIHARA=A [{Q{ ,ut}- iPHt,Pu,}},{zj9,}]

where Pui Pui .Xh. Because the set [{z1,z1},-~,{zn,zn}'] forms anorthonormal basis for H, we have

{ F H t , Pu,} = S5On the other hand, there exists a separation constant C2, which is inde-pendent of A, such that

iV(/-WI0 , ^veobtain1/3-11 111{,f}-/9{i,2i}I I I ,

since |||{ut,ut} ||[= \ut\\=1. Therefore, we have I{ut,Ui} - {zt,zt}1 < I I I{it,,M,}-${z{, z(}I + U I / 9 {zt,zt} - {zt,z{} H

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F E M F O R E IG E N V A L U E P R OB L E M 57Ik-sj lk-sj tf.

This completes the proof.3. Numerical Examples

For the purpose of checking the numerical accuracy of the finiteelement schemes, we have dealt with two examples of the rectangulardomain (LrXL2/ ) . The first example is the following equation:

An+kn0 in &u= 0 on S (Dirichlet type).

The exact eigenvalues and the corresponding eigenfunctioiis are given by

. mty= sm -- sin -Ly

As the second example, we have dealt with the room acoustic w^aveequation for steady state confined by a rigid wall. The equation is thefollowing form:

.= na

0 on S (Neumann type)where P is the sound pressure, c is the sound speed and a)is the angularfrequency. The acoustic analysis is to determine the normal frequencyand the corresponding eigenfunction P. The exact normal frequencyvalues f and the corresponding eigenfunctioiis P are given by

7 , nny / r, -, 0 \p =Cos -cos - , (m, n= 0, 1, 2, ) -Lx LyTable 1, 2 and Figure 2, 3 show the results of the computations incomparision with the exact eigenvalues and the normal eigenfrequencyvalues (in Hz) for a 2.0x1.1 rectangular domain. They demonstrate

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58 K A Z U O I S HI HA R Athat the eigenvalues and normal frequency values converge with the meshsize and the CM scheme gives th e upper bound and the LM schemegives the lower bound for the exact values. And the M M scheme givesmore accurate approximations between the CM scheme and the L Mscheme.

W e have used FACOM 230-28 computer in Ehime University andFACOM 230-75 in Kyoto U niversi ty .

Remark. A s a natural extension of the M M scheme, w e canpro-pose th e generalized mixed mass (GMM) scheme wi th a parameter a(0

Table 1 Th e results fo r Dirichlet type.Mesh Configuration

(Size: 2.0X1.1)

1

2

3

//s'//''///S//s^///////' S' S* //

/*"/S*sS' sS'

' S' s' /' /''//S/*//'/s'S*'/S~ ? s

SchemeCMM MLMCMM ML MCMM ML M

Exact

Eigenvalues(1,1) M ode

13.6611.349 .69

12.3111.0910.0911.6910.9410.2810.62

(1 ,2) Mode30.6219.4414.1925.2919.4415.7522.6119.1216.5318.03

(3,1) Mode54.6433.7524.5646.8930.1721.4042.5731.4024.2530.36

654

LM1 2 3 mesh 1 2 3 mesh 1 2 3 mesh

(a) (1,1) mode (b) (1,2) mode (c) (3,1) modeFigure 2 Convergence of the eigenvalues (Dirichlet type)

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F E M F O R E I G E N V A L U E P R O B L E M 59Table 2. The results for N eumann type .

Mesh Configuration(Size: 2.0 Xl.lm)

1

2

3

4

\s/\/r/// < /L7/'//////~ 7/ss< / _/ ////"//>

^ /^^ /^^ ( ^s'

^ _ ,/ _ ,/ ,_ / _s ^s?Sf/ / _,/l?tX./s s

// /s//s/*/

Q1bchemeC MM MLMCMM ML MCMM MLMC MM MLM

Exact

Hz'100

95,9 0 1 - ,60 18075'

1

M^/IM exa-e

Hz170160150

ct 14C1 3C1 2C1 1 Ci n r

Normal Frequencies (Hz)(1,0) Mode

93.183.375.888.784.681.087.184.982.886.485.083.685.0

^*""-*- s exact7/J

)/^

(0 ,1) Mode167.6131.6107.0160.9153.6138.6158.2154.2150.4157.0154.4151.9154.5

H z20 0190180170160150140

2 3 4 mesh 1 2 3 4 mesha) (1,0)mode ( b ) (0,1)mode

(2 , 0 ) Mode187.5151.5140.3195.6160.8146.8186.0166.9151.2180.7168.7158.3170.0

exact

'{1 2 3 4 mesh C ) 2 , 0 ) mode

Figure 3. Convergence of the normal frequencies (N eumann type)

The GMM scheme includes as its special cases(i) CM scheme (a = l),( i i ) MM scheme (a.=1/2),(iii) LM scheme (a' = 0).

We can p rove the convergence o f t he G M M scheme using th e sametechniques used for the M M scheme. The proof wi l l be published ina for thcoming paper .

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60 KAZU O ISHIHARAAcknowledgements

The author would like to thank Professor M . Yamagut i of KyotoU niversi ty and P rofessor T. Yamam oto of E hime U niversity for theirvalua ble suggestions and continuous encouragem ent. A lso he is gr atefulto the referee for comments and for suggesting improvements in the pres-entation.

References[ 1 ] Fujii, H., Finite element Galerkin method formixed initial-boundary value problems

in elasticity theory, Tech. Rep. CNA-34 Center for Numerical Analysis, The Uni-versity of Texas at Austin (1971).

[2] Fujii, H., Finite element schemes: stability and convergence, Advances in Compu-tational Methods in Structural Mechanics and Design, UAH Press (1972),201-218.

[ 3 ] Strang, G. and Fix, G., An analysis of the finite element metod, Prentice-Hall, 1973.[ 4 ] Weinstein, A. and Stenger, W., Methods of intermediate problems for eigenvalues,

theory and ramifications, Academic Press, 1972.[ 5 ] Washizu, K., Some remarks on basic theory for finite element method, Recent

Advances in Matrix Methods of Structural Analysis and Design, UniversityofAlabama Press (1971), 25-47.