..

II

UARI Research Report No. 129AFOSR Scientific ReportProject THEMIS AFOSR-TR-73-0020

CONVERGENCE. ACCURACY. AND STABILITY OF FINITE-ELEMENT APPROXIMATIONS

OF A CLASS OF NONLINEAR HYPERBOLIC EQUATIONS

by

J. T. Oden and R. B. Fast

Research Sponsored by Air Force Office of Scientific ResearchOffice of Aerospace Research. United States Air Force

Contract No. F44620-69-C-0124

The University of Alabama in HuntsvilleSchool of Graduate Studies and Research

Research InstituteHuntsville. Alabama

August 1972

Approved for public release; distribution unlimited

CONVERGENCE, ACCURACY, AND STABILITY OF FINITE-ELEMENT APPROXIMATIONS

OF A CLASS OF NONLINEAR HYPERBOLIC EQUATIONS

J. T. Oden* and R. B. F08tt

Department of Engineering MechanicsThe University of Alabama in Huntsville

SUMMARY

Numerical stability criteria and rates of convergence are derived for

finite-element approximations of the nonlinear wave equation Utt - F(ux) ;

f(x,t), where F(u ) possesses properties generally encountered in nonlinearx

elasticity. Piecewise linear finite-element approximations in x and cen-

tral difference approximations in t are studied.

1. INTRODUCTION

This paper is concerned with the estimation of the numerical stability

and rate-of-convergence of finite-element approximations of transient

solutions of a rather wide class of nonlinear hyperbolic partial differen-

tial equations. The equations of motion of practically all homogeneous

hyperelastic bodies, including both physically nonlinear materials and

finite amplitude motions, fall within the general class of equations con-

sidered here; but, for simplicity, we limit ourselves to one-dimensional

bodies (i.e., one dimensional spatial domains) and we assume that the

initial data and the solution are smooth functions of x and t. Thus,

*Professor of Engineering Mechanics, The University of Alahama in Huntsville.

tGraduate Student, Georgia Institute of TechnOlogy, and Senior ResearchAssistant, The University of Alabama in Huntsville.

.r

"'.

-I ,

,"

t"'.

t

.

2

while we rule out sharp discontinuities such as shocks, our results are

perfect ly va lid for mos l tl"ansient non Iinear vibration proh l~ms. The

essential featun.· of the pncsent analysis is that full discretization in

both space (Le., in the particle labels x) and in time t is used: the

finite elel11cntmethod 10 used to approximate the variation of the solutioo

in x, whill' ordin:lry central differences are used to approximatl' various

time -ra tes-of ch:mge.

The study of accuracy :.Indconvergence of finite-element approximatia1 R

hus, until recent times, heen largely confined to linear strongly elliptic

operators (see. for eX:lmple, [1-9]). Generalizations of certain results

to classes of nonlinear elliptic-type problems have been discussed by

Ciarlet, Schultz, and Varga [10-13], Melkes [l4], Oden [l5,16J and Varga

[IlJ, but extensions to problems of the evolution type have come about much

more slowly. The work of Douglas and Dupont [l8J provides a basis for

deriving error estimates for linear and certain nonlinear parabolic equa-

tions, and Kikuchi and Ando [19] presented a penetrating study of proper-

ties of finite-element approximations of a class of linear and nonlinear

equations of evolution. Fix and Nassif investigated finite-element approxi-

mations 01' certain linear parabo Iic cquntions [20] and linear rirot-order

hyperhoUc equations [2l]. More recently, FuJii [22J examined tile sUJhility

,lOd convergence of finite-element approximations of smooth solutions or

linear second-order hyperbolic equations in which Newmark's S-method is

used to represent the behavior in time. Like Fujii, we also examine

stability in certain natural energy norms; our error estimates essentially

agree with those Fujii obtained for the linear case, but our approach is

necessarily quite different.

We confine our attention to a class of nonlinear wave equations of a

furm very common in continuum mechanics, in which the wave speed is a bounded,

,.

3

continuous, ;)nd always positive function of the gradient Ux of the dependent

variable u(x,t), and for which u(x,t) has continuous third derivatives

,.,rith respect to time t. '111e temporal behavior of u(x,t) is approxinwted

using standard central diffl'rences and the spatia I hehavior is c1l'scriht'd

using piecl·\.,ri se line;] r finite-l'Lement :Ipproximat 10nl>. Within the fr:lllwwork

of these :ISSlllllptions, Wl' show that

. The sLlhi lily of the schl'nw in energy is (]osured if (h/lIt) >

\I C"I) (u )//2, where h io the minimum mesh length for thc finite-clemenlry max T.

model, v (ry = l,2) are constants, \I corresponding to a cl)nsistl~nt 1lJ:l8Sry 1

formulation and \I.... to a lumped mass formulation, and 61) (II) is the maximum.., max x

speed of propagation of acceleration waves relative to the material at the ith

time increment. Obviously, this remarkably simple result reduces to similar

criteria obtained for linear difference approximations when C ;constant.max

. While our stability criteria give only sufficient conditions for stahil-

ity,they suggest that for a fixed h it is sufficient to use a smaller time step

for the consistent ttk'1SS formulation than for the lumped mass formulation since

\11 > "2! (Moreover, use of lumped masses avoids ringing in front of a wave

front and maintains the finite character of wave speeds in the model.)

. Under thE' stated assumptions, the square of the ~ -norm, lIe\~IIl~, of

the gradient of the error at each time step i is O(h~ + (6t)~). (Similar

accuracies are obtained after R time steps in the linear case; cf. [22] .)

Uniform convergence of the error e is also obtained.

. The same rates-of-convergence for the consistent mnss [ormul:]tlon

are obtained for the lumped mass formulation.

4

20 STATEMENT OF TIlE PROBLEM

We prefer to describe the problem in physical terms. Consider a long,

thin homogeneous rod of hyperelastic material of length L, undeformcd

cross-sectional area A , and initial mass density p. We wish to studyo 0

the longitudinal motion of the rod relative to a reference cunfiguration

in which the rod is at rest prior to t = O. In the reference configuration,

particles are labeled by the axial coordinate x, and the displacem~nt of x

at time t is denoted u(x,t). Response of the rod is initiated by some

prescrihed initial velocity v (x) or initial displacement u (x); for theo 0

moment, no hody forces or end tractions are prescriht'd.

The total e,lergy R(t) of thL' rod at time t is given by

E K + 1\ (2. 1)

where K is the kinetic clwrgy :IntiH ls the internD J energy:

H

L

A. JWdx

o

(2.2)

Here II = f-IlI/<ll is the particle velocity field and W is the strain energy

per lInLL unucformed volume. The Htrain energy function W depends upon

dU(X,L)/,)X IIx and is :.Isoumedhereinafter to have the following properties:

(i) W(u ) hal> continuous hounded positive second derivatives withx

respect to tlw displacement gradients ux' Indeed, d~W /du2 is proportiona 1x

..to the square of the natural wave speed of the matel i:.ll,n positive [uncti on

always < -t=.

(li)dW(ux)

du xfirst Piola-Kirchhoff stress (2.3)

5

Property (i) is directly akin to material stability and is :l constitutive

assumption; property (ii) is merely a definition.

Since the SystCll1 it; conservative and the mechanical power of the

external forces is zero, the principle of conservntion of energy asserts

tha t [or every t

E(t) o (2.4 )

Denoting the inner-product of any two displacement fields ul (x, t), u:,>(x, t)

by

L

(", (',t), ",(.,t» =f "''',ctxo

we find, upon introducing (2.2) :lnd (2.3) into (2.4), th:lt

p 1\ (,i, u) + A (T (li ),t'l ) ; 0o () 0 x x

(2.5)

(2.6)

This result is the governing cC(u:ltion for generalized motion!! of the rod;

it is precisely the weak form of the nonlinear hyperbolic momentum equation

PoAo (dT(lIX») (}~~= 0

- A du oxo x(2.7)

Obvii.ous ly, -=- <IT/ duP x

o

of the rod nlateria 1.

= L d2W/du::> is the square of the natura 1 wave speedp xo

We must, of course, add to (2.6) appropriate bOllndary-

.1ml initial conditions of the type (\1(',0) - 1I ,v(·,O» = 0, (il("O) -o

Lv0 ' v ( • , 0» = (), T ( U ) u I = (~ l' t c •

x • 0

:3. FINITE ELa-l ENT/DIFFrmENCE APPROXIMATIONS

To approximate (2.6), we follow the usual procedure of partitioning

the rod into a finite number E of segments connected at nodal points at

each of their ends. The E + 1 nodal points are located at 0 = xo, Xl ,X".., ....

L and the length of an element between nodes xN-l and xN is

d~noted h~ (i.e. h~ xN - xN-1). The exact solution of (2.6) durin~ a

time interval 0 -: l <; T is an element of the space H1(O,L) X (O,Tl of

functionfl whose p:lrtial derivatives :lre squ[lre integrahle on (0, L). In

the finite-element method, we seek an approximate solution in a finitc-

dimensional suhspace of 1I1(O,L), the clements of which :.Ire of tll(~ form

U(x, t) = u~(t)W'-l (x) (3. L)

where the repeated nodal index N is summed from 0 to E + 1. Here U (t)'01

is the value of U(x,t) at node N [It time t and W (x) arc basis functions'"

designed so as to have the usual finite-element properties; i.~.~

ljI (xM) = 0"1 , M, N = O,l, .•• ,Jo: + J

'" '" 0.2)

ljI (x) = 0 if x l (xN-1,XN+1)

'"In general, the [unc tions ljI (x) <l re ~ene ra ted from Joe <)1 (eleml!l1t) hasis

'"functions which contain complete polynuminls of degree p, where p + L

is the highest spatial derivative that <)ppears in the functional (I..h) (see.

for example, (151). In our case, p = 1.

In the present study, we slwll also follow the customary procedure

and approximate the hehavior of U(x,l) in time hy finite differences.

We dividl' the tillle interv[l) [0,'1'1 into R equal lime Intcrvnls C1t. and

descrilw the v;"Iriation of U (t) in t in terms of its vnlul's ;"It tillll'S i~L,'-I .

i = 0, I, ... ,R. Thuo 0.1) is written

Ubi = U (x i6t) = U\llll' (x), " N' '" 0. '3)

where t/Jl = U (i6t). To represent time-rates, we employ the difference'" "

quotients

b. U~) = (uUl - IIi -1) / t\tt

= (U(l+1) - tf1_11)/2b.t

IVt lfll = (Uti + 1) - TJl) / b. t

/) UUI = 1. (IV ull} + b. u(11)t 2 t t

/) ~il = IV~llt Ifii) = (lP + 1) - 2U~)+ Oi -1\) /6t2(3.4)

7

By direct substitution of (:l.]) and 0.4) for u and its various time

rates in (2.6), we [Irrive at the finite-element/difference scheme governing

our discrete model:

P A (IPu&1 6 U(1)) + A (T (Utll) 6 IfII) = aoot't 0 x'tX 0.5)

Here If'x

Physical Interpretation of Equation 0.5). It is enlightening to

interpret the terms in (3.5) physically. For instance, consider the first

term in (3.5), and denote by K1) and ~) the kinetic energy of the discrete

model computed using forward and backward difference approximations of the

velocities, respectively. Then

1 lpl un'(3.6)

~1) = - m \1 "I 'V t ,~+ 2 ~"., t

- .!. 6 U(l)6 tf.1l~ - 2 m'oJ'I t '.l t ,~

where N, M = 0, 1 , .•. , E + 1; i

nwsS matrix,

1,2, ••• ,R - land m is the consistent"I'"

1.

P A f $ tI, .. dxo 0 .. ~

o

(3.7)

It can now be shown that the first term in (3.5) is, as should be expected,

precisely.§!. difference analogue of the time rate-of-change of the kinetic

energy £.L the finite-element model. In fact,

1= - m (\1 tflJ\7 U(1) - t!1 UIl'1\ ulJJ12t!1t "11.\ t "I t 1.\ t "I t 1.\

(J.8)

8

Note olso that if Ilull denotes the norm associated with the inner product

(2.5) (Le. llull;~ = (ll,u», then 0.6) can also he written

(3.9)

TIH~ second term in (3.5) is clearly analogous to the time rate-of-change

of the total internol energy or the model:

H A (\~, 1) = Ao (ddW , tl ) .~ A (T(tjU') ,Otlfll)o u x 0 x .xx

(3.10)

This term contains the st if fness re lations for the finite-element mode l.

In f:lct, inlhe lineor theory T(l~xh) = mU1 = If!IE\~ , E heing the claslicx N N , X

modulus, so that H :::::J A itfllo tfI!(\~ ,~.. ); the array K = A E(I~ ,~ )o "I t r" "I , x ~, X >J M 0 N ' X ~ , x

is the global stiffness matrix for the model. In the present study, of

course, the stiffness relations may be highly nonlinear.

Relations (3.8) and Cl.IO) indicate the manner in which 0.5) approxi-

mates (2.6). Equations (3.5) describe a discrete model of the conservation

principle (2.4). Since the quantities 0 utI) (i = 1,2, ... ,R - 1; N = 0,1, ... ,t >J

E + 1) are linearly independent, it is clear that (3.5) also implies the

discrete momentum equation,

(3.11)

However, we prefer to retain the l'lwrgy form ().5) for reasons which wi LI

hecome apparent suhsl'quently.

4. NUMERICAL STABILITY

We now investigate the stability, in an energy setting, of the nonlinear

finite-element/difference scheme (3.5). To interpret energy criteria in

physical terms, we observe that for the continuous system, (2.4) implies

that

•TJ ;''''lo

I':(T) - ,·;(0) o

9

(4. l)

While we cannot expect the discrete model to also hehave in this ideal

fashion owing to round-off errors inherent in our method, we can expect

that errors in the energy do not hecomc unbounded during the time interval

(O,TJ. In other words, if Eh(t) is our finite clement approximation of the

total energy at time t, we slHlll consider our numerical scheme to be

stable in energy, if there exists a constant C > 0 such that

for i 1,2, ... ,R

Equivalently, stability in energy is assured if

(4.2)

for all i = O. I, 2, ...•R.

Our stahi Ii ty mwl ysi s is ha seu on the £ollowing assumptions:

• W(lI ) has property (1) 0 f (2.3).x

The finite-eLemeTlf interpoLation functions. (x) satisfy the usualN

convergence illlli c.:()mpll'tclll'S~;criteria for l.1nl'urelliptic probll'ms; i.e.,

the family of functions UNWN(X) cont:dlls complete polynomiul.s of degree

p = 1 and the finite cl.ement approximation U • (x) is continumls acrossN N

inlcrelement buundaries. In particular, • (x) may he taken to he the'"

common piecewise lincar pyramid functions

o

[h + (x - x )]/hM ~ ~

(X - x) /hM+1 ",+1

x , [x"'_l' XM~l 1x € [X"_l' x",]

x € [xM' XM+1] (4.3)

We next cit l' the fo1lowing lemmas:

10

Lemma 1. Let (4.3) hold, and denote Q..y h the minimum element length

h = min h"I

N=l.,E+J

Then, [or ~very finite element approximation U

'VI

II/JUII S; - IlulIox h

where lIull:-> = (U, U) and \/1 is the conI;;tant 2/3.

(4.4)

Incqu:llity (4.5) is :.I fairly well known result :.Ind can be found else-

where (e.g. [22,23]). It can be proved directly by merely substituting

(4.3) into (4.5) and carrying out the integration. We omit the details

here.

Lemma 2. Let the strain energy function W(ux) satisfy property (2.3).

Then

where

(4.6)

lJ41x

UOl + e (If! + 1) - ttll)x x x

(4.7)

Proof: If (2.3) holds, we may expand the strain energy function in

a finite Taylor series about W(l) = W(U~l) (this is an implicit expansion

in time):

TiP +1}

IJ. -1) \J'l - T(~\)(IP\ - l.P--U) +1 T'(~l)(~\ - 1~-1l)C1x x 11 2 x x x

(4.8)

(4.9)

11

where T(ux) '" dW/dux and T' (ux) = d2W/du~ = dT/dulr• Subtracting (4.9)

from (4.8), multiplying the result by Ao/2/).t, and integrating over the

length of the bar gives the desired relation.

Our numerical stability criterion is given III

Theorem 1. Let Lennnas 1 and 2 hold. Then a sufficient condition- - - --for the finite-element/difference scheme defined ~ (3.5) to be stable

in energy in the sense of (4.2) ~ that

\)1

!.!..... > - e(l)M /2 max

(4.10)

where \)1= 2(3 and C~~x is the maximum~ !p'(~ed experienced ~ the rod

for all x at time t = i~t:

Proof:

~= max[T I (tPi) / Po Jx

According to (3.8) and (4.6), we have

(4.11)

(4.12)

where H~- t~ = i [(ll'- +1) + I-tll) - (tfJ> + I-fj -1) J = ~ (~+u - ~ -1) • Observing

that T' (ux) is always positive, that \I~+1) - ~\\2 ~ ~t2'V\\IVtrP)\I2/h2, and using

(3.9), we find, after rearranging terms, that

(4.13)

Now ~~+~ = ~~-1) + l~-l). Moreover, if there exists a positive constant

a such that 0 < a ~ 1 and such that

12

(4.14 )

2then obviously afl(l1l + If':)) ~ I(tl)(l- ~t2\)2C(1)12h2) +~) ::;; ~-l)+i-P.-l). Our\ + + + 1 max + + +

stability criterion comes from the fact that (4.14) is satisfied if

~t2 \)fC(l) ::;; 1 - a < 1

2h2 max

which leads directly to the desired result, (4.10).

(4. 15)

This stability estimate, as should be expected, is consistent with

the well-known von Neumann linear stability criteria [26J which requires

the discrete system to propagate information at a rate greater than or

equal to the speed of propagation of the actual system. The stability

criterion (4.10) for the nonlinear system (3.5) simply requires the

approximate wave speed to exceed the actual wave speed times a constant,

(\)1(2) ~ 1, for all x € (O,L). The constant \)112 depends upon properties

of the discrete model: by LemmB 1, (\)1112 a 16 for A consistent mass llpproxi-

m:.ltion,and, as discussed in the following remarks, (\)112) = 12 for

the lumped mass approximation.

REMARK 1: If a lumped-mass finite-element formulation is used ratre r than

a consistent-mass formulation (i.e., if mNM is diagonal), then the constant

V1

in (4.5) must be replaced by ~ = 2. All other steps in the :lnalysis

are the same. Consequently, instead of (4.10), we obtain stability for

the lumped mass model if

!:!.- > \)2 ( 16t -r.:>2 C) = 12 C(O1/ ~ Illax

REMARK 2: Our stability criterion (4.20) is not altered if

(4.16 )

we consider, instead of (2.6), the nonhomogeneous wave equation, p A (u ,v) +o 0 t t

A (T(ux»)v )=Ao(f,v), provided 1If(·)t)11<<:Xl for all t. The inclusion ofo x

13

such a nonhorno¥eneous tI.'rmmerely adds to (4.1) a term on the ri~ht side

of the form c S 1If(·,t)II"dt.

oREMARK 3: The stability criteria is easily adapted to variable step time

integration schemes, since it holds "step-wise"; i.e., it is developed for

a typical time step t:lt. Thus, a possibly more useful criteria is obtained

by replacing titin (4.10) by (tlt)\1l.

5. ERROR ESTIMATES AND CONVERGENCE

Our error estimate and consequently Ollr converg~nce proof depend

strongly on the smoothness properties that u(x,t) and W(u ) an- assumed tox

have. The strain energy function W(u ) is assumed to have the houndedx

positive character reflected in (2.3). Indeed, we noted earlier that

d2W /du2 = T' (u ) is a constant times the squared speed of propagation ofx x

simple waves for the material. It is customary to assume, for real wave

speeds, that this function is always positive and, barring such exceptional

caSeS as perfectly constrained incompressible materials, is also finite.

If T(u ) = dW/du is continuous in u , then, for any two displacementx x x

gradients u x unJ w , T(u ) = T(w ) ± T'(w + A (ux- w »(u - w ), Ax x x x ~ x x x ~

e ,e , 0 ~ e ,0~ ~ 1. Hence1? 1·

c (u - w ,v ) ~ (T(u ) - T(w ),v ) ~ c (u - w ,v )oX xx x x x 1 x x x (5. L)

c o (5.2)

Moreover, at points (x,t) which do not lie on the surfaces of discontinuity

corresponding to acceleration waves, the exact solution u(x,t) can be

assumed to have third derivatives with respect to time. Therefore,

J:~ ( '''t) = ~u(x,il.it) + lIv t)"t\)t.ll x,~(..l ot2 'v \X, '.1

14

(5.3)

where ,.JIl(x,t) = [li (x,i(1 + Cl)lit) -'cr (x,i(.l - e)lit)J/3~, with 0 s; e, Ii s; I

and IlrJlll1 < rl>. These observations and assumptions set the stage for our

convergence study.

We consider now the nonhomogeneous form of (3.5); i.e., if u(x,t) is

the exact (generalized) solution and U*<1l= U*(x,ilit) is the finite-elemert:

solution approximating u(x,t), thcn

and

(f, V) + P (eJIlf\t, V)o

(5.4)

(5.5)

Here I\fll < (X) and V = V~1jIN is an arbitrary clement of the finite-element

subspace Ill.. Moreover, if lfll = U(x, il.it) = U~~~ (x) is :.Inother arbitrary

function in m, a little algebra leads to the relation,

(5.6)

Now, ~s is customary III cUHtomary in convergence studies of lincar

elliptic problems, we shall identify lill in (5.6) BH the [lnite-element

interpolant U(ll of the exact solution lfll• That is, if u(x, l) is ~iven,

ij(x,t) is that element of the finite-clement subspace m that coincides

with u(x,t) at each nodal point xN for every il.it, i = O, .•. ,R. It is well

known (see, for example [151, pp. 1ll-lI6) that if U is a linear combination

of the functions 1\1 (x) of (4.3), thenN

and (5 n 7)

15

Here µo :Jnd P'l are pOoitive constcmts, ~l) ::: ifll - Ifi), and ~\ := ~X(fJCi) - d,1»).

The qu:mtity 1·~1l is the interpolation error at t = i~t, whereas the actual

error inhcrl'llt in tlw finite-clement solution is

(5.8)

where

t~1!l = ift) - U*(n

The stage is now set for the following lemma.

Lemma3. Let (S.l) - (5.6) hold, Then

p (62&) en) + r.t 11r;1l112 ~ P (62~) e<tl) + ry h:? + ex (~t)2at' ox ot' 1 :il

and

where ~ , 8 , r = 0, 1, 2 are positive constants.r r -

(5.l0)

(5.11)

Proof: While somewhat lengthy, the proof is str:.lightforw:Jrd. We shall

only out line the essentia I steps. Our proof makeH usc of the inequality

(u,v) ~ ~€ \lu\l2 + ~lIv\l2 , € > 0 (5.12)

which follows from the Schwarz inequality and the elementary inequality

labl ~ (1/4€)a2 + €b2, and the fact that there exists a constant k > 0

such that

I\u\I ~ k\lux\I (5.13)

Inequality (5.13) is the well-known Friedrichs inequality And its validity is

pmved in a nunberof texts (e.g.,[24],p.290). 1\> obtain (S.lO),set V=e(t)in (5.6),replncl'

16

the second Lerm on the left side hy cllle5~III::> using (5.1), and apply (5.12)

:md (5.l) simul t<lneous ly to the second and third terms on the right side.

Then making usc of (5.7) and (5.13) and collecting terms giveo the desired

result. Equation (5.ll) is obtained by subtracting (5.4) from (5.5),

uoing (S.l), applying inequalities (5.12; and (5.13), and thti!n using the

identity (5.8). This completes the proof. We now have

Theorem 2_. Let (s.l) - (5.6) hold. Then, ~ h and i.\t -+ 0, dO in the

finite-element approximation satisfies

(5.14)

where Mo and M1 are constants independent of hand i.\t.

Proof: We know from the triangle inequality that

(5.15)

The term Ilr~~111is O(h) in accordance with (5.7). To e1:itimate lIe\~\II, W(.' ohserve

that (5.10) implies that

Since r11l - e(\l = et\), we may introduce (5.11) to ohtain

where Yo' '1'1, :.Ind yare constants. Substituting this result along with:J

(5.7) into (S.lS) gives (5.14).

Equation (5.14) proves that the finite-element approximation UoJC'(11 con-x

verges in the mean to the gradient of the exact solution lPI for each i.x

We assume, of course, that in taking the limit h, 6t -+ 0, the stability

criterion (4.10) is satisfied.

17

Since lIeU11I S; C Ilell)ll, convergence of U-M.1)tox

Jtl is also assured, but we do not attempt to assess its rate of

convergence here. Since the Sobolev inequa lity sup I e(1) I s; Clle(t) II +

Ild!)") holds, we observe, as did Fujii [22J, that U.;.<n converges uniformlyx

We also remark that the use of lumped or consistent masses has

no bearing on the final error estim<t:e, other than possibly altering the

constants No ilnd M1 in (5.14): the lumped mass and consistent muss ;lpproxi-

mations have the same ratc-or-convergence.

As a final comment, we remark that while our results are limited to

rather opecific classes of approximations of both the temporal operators

and the spatial approximations, the approach is quite general. An investi-

gation of a number of generalizations of the problems discussed here shall

be the subject of future work.

ACKNOWLEDGEMh~T.We arc indebted to Drs. H. Fujii and George Fix fortheir comments on an early draft of this paper. The support of this workby the U. S. Air Force Office of Scientific Research under ContractF44620-69-C;-OI24 is gratefully acknowledged.

7 . REFERENCES

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CONVERGENCE) ACCURACY. AND STABILITY OF FINITE-ELEMENT APPROXIMATIONS OF A CLASSOF NONLINEAR HYPERBOLIC EQUATIONS

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Numerical stability criteria and rates of convergence are derived forfinite-element approximations of the nonlinear wave equation u' - F(u ) =f(x.t). where F(ux) possesses properties generally encounteredtrn nonlInearelasticity. Piecewise linear finite-element approximations in x and centraldifference approximations in t are studied.

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