..
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
1. E. R. A. Oliveira, "Theoretical Foundations of the Finite ElementMethod," International Journal of Solids and Structures, 4, 929-452(1968) .
2. N. \01. Johnson, Jr. and R. W. McLay, "Convergence of the Finite Eh~mentNethod in the Theory of Elasticity," Journal of Applied t-lechanics,Series E, 35, No.2, 274-278 (l968).
3. M. Zlamal,-"On the Finite Element Method," Numeriche Mathematik, 12,394-409 (1968).
4. G. Fix and G. Strang, "Fourier Ana lysis of the Finite Element Methodin Ritz-Galerkin Theory," Studies in Applied Mathematics, 48, 265-273(1969) •
5. I. Babuska, "Error-Bounds for Finite Element Method," NumericheMathematik, 6, 322-333 (l97l).
6. I. Babuska, iTThe Rate of Convergence for the Finite Element Method,"SIAM Journal of Numerical Analysis, 8, No.2, 304-315 (l971).
7. 1. Babuska, "The Finite Element Nethod for Elliptic Differentia 1Equations," Numerical Solution of Partial Differential Equations - II,SYNSPADE1970, Edited by n. Huhhard, Academic Press, New York, fllJ-I06,1971.
18
8. G. Strang, "The Finite Element Method and Approximation Theory,"Numerical Solution of P:.IrtialDifferential Equations - II, SYNSPADE1970, Edited by B. Hubhard, Academic Press, New York, 547-584, '·1971.
9. G. Strang and G. Fix, An Analysis of the Finite Element Method,Prentic-Hall Englewood Cliffs, New Jersey (to 3ppear).
10. P. G. Ciarlet, M. H. Schultz, and R. S. Varga, "Numerical Methods ofHigher-Order Accuracy for Nonlinear Boundary-Value Problems. I. One-dimensional Problem," Numeriche Mathematik,?_, 394-430 (1967).
11. P. G. Ciarlet, M. H. Schultz, and R. S. Varga, "Numelt'icalMethods 'ofHigher-Order Accuracy for Nonlinear Boundary-Value Problems. II.Nonlinear Boundary Conditions," Ibid, 11, 331-345 (1968).
12. P. G. Ciarlet, M. n. Schultz, and R. S.-Varga, "Numerical Methods ofHigher-Order Accuracy for Nonlinear Boundary-Value Problems. III.Eigenvalue Problems," Ibid, l2, l20-133 (l968).
l3. P. G. Ciarlet, M. H. Schultz;-and R. S. Varga, "Numerical Methods ofHigher-Order Accuracy for Nonlinear Boundary-Value Problems. IV.Periodic Boundary Conditions," Ihid, 12, 266-279 (1968).
l4. F. Melkes, "The Finite Element Method for Nonlinear Problems,"Aplikace Matematiky, Svazek l5, 177-189 (l970).
15. J. T. Oden, Finite Elements of Nonlinear Continua, McGraw Hill BookCompany, New York, 1972.
16. J. T. Oden, "Finite Element Models of Nonlinear Operator Equations,"Proceedings, Third Conference on Matrix Methods in Structural Mechanics,Wright-Patterson AFB, Dayton, 1971.
17. R. S. Varga, Functional Analysis and Approxinllltion Theory in NumericalAnalysis, SIAM Regional Conference Series in Applied Mathematics, SIAM,Philadelphia, 1971.
lB. J. Douglas and T. Dupont, "Galerkin Methods for Parabolic Equations,"SIAM Journal of Numerical Analysis, 7, 575-626 (1970).
19. F. Kikuchi and Y. Ando, "A Finite Inement Method for Initial ValueProblems," Proceedings, Third Conference on Matrix Methods in Struc-tural Mechanics, Wright Patterson AFB, Dayton, 1971.
20. G. Fix and N. Nassif, "Error Bounds for Derivatives and DifferenceQuotients for Finite Element Approximation of Parabolic Equations,"Numeriche Mathematik (to appear).
21. G. Fix and N. Nassif, "On Finite Element Approximations to Tinle-Dependent Problems," Numeriche Mathematik (to appear).
22. H. Fujii, "Finite Element Schemes: Stability and Convergence,"Advances in Computational Methods in Structural Mechanics and Design,Edited by J. T. Oden, R. W. Clough, and Y. Yamamoto, The Universityof Alabama in Huntsville Press, Huntsville, 1972.
23. 1. Fried, "Accuracy of Finite-Element Eigenprohlems," Journal of Soundand Vibration, 18, 289-295 (l971).
24. S. G. Mikhlin, ~athematical Physics, An Advanced Course, North-HollandPublishing Co., Amsterdam, 1970.
25. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,Interscience, New York, 1948.
26. G. G. O'Brien, M. A. Hyman and S. Kaplan, "A Study of the NumericalSolution of Partial Differential Equations," Journal of Mathematicsand Physics, ~, 223-251 (l951).
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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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