51..\\1 J. 'lATH. ""'·\1Vol. 12. .slJ n. ~i"l\t"m"t"r 19H1

i lq)\) ,.I~'ll·l\fm Indu ..tnal Jnd Apphed ~f;~lh~m,;ul('1.\.n~n-l-llllfJll:'l::~-U01: SOlIN! II

ANALYSIS OF GALERKIN APPROXIMATIONS OF A CLASS OFPSEUDOMONOTONE DIFFUSION PROBLEMS*

G. ALDUNCIN7 AND J. T. ODEW

Abstract. A class of nonlinear parabolic problems characterized by convective terms which dependnonlinearly on the solution and its gradient. are considered. Specifically. the operators characterizing theproblems are shown 10 be pseudo monotone and to satisfy G:hding inequalities. The existence. uniqueness.and Galerkin and Faedo-Galerkin approximations of the general class of nonlinear diffusion problems areinvestigated.

1. Introduction. In this paper we are concerned with the existence. uniqueness.and Galerkin and Faedo-Galerkin approximations of the following general class ofnonlinear diffusion problems: Let 0 be a bounded domain in (R" with boundary ao.and 0< T < 00. Given data f in 0 x (0, n and initial data 1111 on 0, find II =ulx, t). (x, f) EO x (0, n, such that

au--v·a(Vu)+b(lI.vII)=f inQ=Ox(O. n.at

11.1 )

where

u = 0 on ~ = aO x (0. n.Ill' ,0) = Uu on O.

( 1.2)

alv II) = aY' u + k Iv1I11'-~V'II.

a. k E L :>e(m, 2~p <00.a(x)E:;all~() and klx)E:;kll>O. a.e.xeO.

and btu. VII) = b(x, u(x, f), V'U(X, f» is subject to the conditions:

CL

(1.3 ) q = 0 or q ~ 1. r = 0 or r ~ 1,

l~q+r<p-1.

ell. b (t:. t) is totally Frechet differentiable in (R x (R" and its partial derivativesiJ,b: (R x (R" -+ .?(R (R) and iI,b: (R x (R" -+ .?(lRn

• (R) are such that, for (q. r) satisfying (1.3)and 'V«(. t) e (R x (R".

la,b«(, t)1~ cql(lq-1Itlr if q ¢ 0,

la,b(l', t)1~ c,I,lqltr-' if r ¢ O.

The case r = 0 will be understood as b = b(u) (not a function of V' u), and the case q = 0as b = b(vu) (not a function of II).

It is well known that nonlinear diffusion terms such as those represented by theterm a (Vu) in (1.1). are useful in modeling nonlinear heat conduction. They also occurin models of the flow of non-Newtonian fluids. particularly in the study of molten metals

.. Rcceived b)' the editors Junc 21. I 97!!. ancl in final rcviscd form Novcmber 10. 1980. The workreponed here was completed during the course of a projecl supported hy the U.S. Army RcsearchOfficc-Durham under grant DAAG 29o77·G-OO!!7.

t Texas Instiluh: for Computational Mechanics. University of Texas at Auslin. Texas 78712 .

911

918 G. ALDUNCIN AND J. T. ODFN

and in certain problems of flow through porous media. However, it is nO\v widelyrecognized that convection and advection play an important role in many of thesephysical processes, and that adequate mathematical models of such processes shouldfrequently include the effects of nonlinear convective terms. such as bl It, VIII. Thepresence of such convective terms, however. leads to solutions which diffcr consider-ably from those obtained in pure diffusion problems; solutions can exhibit shock-likefronts; uniqueness. stability and regularity of solutions become more important issues.and the analysis of the behavior of approximate solutions is significantly more compli-cated. It is standard practice in studies of Galcrkin approximations of such nonlinearconvection-diffusion problems to restrict the classes of problems under study in sucha way that the methods of monotone operators can be used to obtain error estimatesand theorems on convergence. While such studies are not without somc value, theysidestep the major difficulties mentioned above and may not be applicable to manyproblems of physical interest.

Our objective in this paper is to analyze two types of approximations of classes ofnon monotone parabolic problems of the type (1.1) using the theory ofpseudomonolOne operators. These include fully-discrete Galerkin methods and. undersome additional restrictions, semi-discrete Faedo-Galerkin methods of approximation.We note that, in general. this type of semi-discrete approximation is not necessarilywell-defined for coercive pseudomonotone parabolic problems.

Following this introduction, in § 2, we show that the spatial operator in (1.1) iscoercive and pseudomonotone on a dense continuously embedded subspace of theBanach space LI'(O. T; W.I,"·(O)) and that this implies thaI solutioll~ to (i.l) do exist inL""(O. T: L~(O»nLI'(O. T; Wkl'(O» [6]. In general. multiple solutions will exist to(1.1> and there cannot exist a continuous dependence on the data. However. regularityconditions on the solutions can be given which will guarantee their uniqueness, andthese are discussed in § 3.

In § 4 we introduce an elliptic regularization of problem (1.1) of the type used byLions [5], and describe properties corresponding to Galcrkin approximations and wegive an approximation theorem which establishes their strong convergence. We alsoderive error estimates for such approximations. Finally in § 5 we describe Faedo-Galerkin (semi-discrete) approximations and show that, in the case of our modelproblem. sufficient conditions are satisfied which guarantce the existence and alsouniqueness of these approximations. We also prove sutlicient conditions for weak andstrong convergence of such approximations and establish corresponding approximationerror estimates.

Notation. (V. 11·11> is a real, separable, reflexive Banach space with topological dual(V', 11·11*), (. , .) denotes the duality pairing on V' x V. (H, ( ... ).1·1) is a real Hilbertspace identified with its dual, in which V is densely and continuously embedded:V'-+ H '-+ V'.

(V, 111-111> denotes the usual space LI'(O, T; V), 2 ~ p < 00, which is a scparable.reflexive Banach space, whose dual space can be identified as (V' .111·111* 1=["'(0. T; V'), p' = p/(p - 1). [', .] dcnoting the corresponding duality pairing.LW,(' .. hI", I'I~) denotes the Hilbert space e(O. T; H) which, being identified with itsdual, is such that 'V '-+ :It '-+ r.

(~,III'III'tl) and (W. IIHlu') denote the separable, reflexive Banach spaces

6IJ = {v: v E 'V, li E .W'}.

')',...= {v: l' e 'V, V E r'}.

Illv!II'tl = IIIvIII + IIvll:lt".II/vlllll' = IIIl'lli + III(illl*.

UALERKIS APPROXIMATIOr-;S 919

Here t· = ilt,! iff is the distrihutional time derivative of [' which belongs to2'«(0, T): V)='sp(g((O. nl. VI. Hence Id. [4] and [5]), ~I '-+ 'w ......r 'TV is con-tinuously embedded in 0[0, T]: HI and, if II, t' E W: II, t' satisfy the Green's formula

( 1.4) [Ii. r] = (II( TL t'(TI)-(II(Q). t'(O))-[i'. II].

Moreover. the trace mappings p·.....L'(O) and [:-+t'(T) are such that {L'(0l: t· E "It'} = H ={t" Tl: t· E 'JI1. (toW): t' E '?/} and {L'( Tl: v E 5lL} are dense in H.

2, Existence anallsis. For the model problem 11.1). we take as spaces \l and H.the usual Sobolev spaces

(2.1) V=W/,·I'(OI. 2;ap~00, H=e(O).

Then. problem (1.11 assumes the following ahstract form:Find II E W such that

(2.2)

illl f-+A(II)= •(If

/((01 = 110.

f given in '1"'.

110 given in L ~(O).

12.31

(2.5)

where A: t· -+ Y is defined by

[.11/11. L']=[A1tlll. d+[A:(1I1, d.

[A 1(11). d,;: L a(x. ~/I(x, f)). Vt'(x. tl dx dl.

[A~(II). t'1 = L bL\'.II(X, rl. VII(X. n)L'(x, r) dxdt,

in which a(\II) is as defined in (1.21 and bIll. VII) is subject to conditions CI and ClIoWe now proceed to establish the existence of solutions to problem (2.21. The

following two theorems detcrmine basic properties of the operator A.THEORE\I 2.1. Ln A: Y -+ '1'" be the operator defined in (2.3). Then i) A is

bounded, ii) A is coercive. and iii) A is locally LipschifZ confinllous in the sense fhaf'V u. /' E B" (0) = {to e l': IIIeli I< µ, µ >O},IV e Y. there is a posifive COll5tQllf Clµ\ StIch fhat

12.4) :[A 1111- AI v l. I\'JI ~ ClµllIlu- L'/II' ill 1\'111·Proof. Wc shall use (hc notation

a,.,=llaIlJ,--'III' k.r=lIklle'lOl and IH,.o=IHL·tOI'il Applying Holder's inequality, we easily obtain thai

lilA (dll* ~ax' mes (Q/I'-Zl/l'lIIvlll+ k",lIIvIl/P' 1

+c mes (Q)'I'-1 .q.rl/'lIvll/q·, VI' E r.ij) From Friedrichs' inequality. it follows that, VI' e L' (0. T; wk' (0», 1 ;a s <00,

12.6) 111'11;'111.7': W,','WI' ~ Ic '(5,11.1 mes lmsl.. + I)IIV 1'11:.0.

Thus,

(2.7)lA (d. d~·alJilVl·lIi.o+ko( I +c"( p, II lmcs I !l)"'" ) 11111'111"

-c mesIOl'" I q "/l'lIIvlll,~q" 'VI'E /'.

920 G. ALDUNCIN AND J. T. ODEN

(2.9)

(2.11 )

But, using Young's inequality in the last term in (2.7), leads to

(2.8) [A(v), l']~aoIIVvllto+')'dllvllll'-')'2T 'VVE Y,

where ')'1 and ')'2 are >0. Therefore, [A(v), vl/lllvlll-++oo as IIIdll-+oc.iii) By the inequality in (Rn [10],

Ilxl'-2X -lyl'-2yl <c{lxl + lylr-2lx - yl,c=,,'r-l if2~r~3, c=r-l if3~r<00

and Holder's inequality. we obtain, 'Vu, v E Bµ(O)c '11, IV E '11,

(2.10) I[A.tu)- A Ill'), w]1 ~ {ax mes (0)(1'-2)/1' + kocc(p)(2µ )I'-~}/llu - dUll/will.We now use hypothesis cn. First observe that

f. IIdh({, V{)[A~(II)-A2(v), w]= 0 0 dO wdOdO

1

= L In {a~b({, V ~)l'j + av~b(g, V~) . V l'j}w dO dO.

where g = l' + Ol'j, 17= u - l' and 0 E [0, 1). Hence, because of CII and Holder'sinequality,

I

/[A2(u)-A2(v), w]I~(cq+c,)mcs(Q)II'-I-q-r1II'Jr llI~mq~'-ld811/11!1!l!II'lIp.0.()

Then, since II, v E Bµ (0) c "/1', there is a constant ')'3 = ')'3 ( p. q, r, 0) such that

(2.12)

(2.13)

Therefore, (2.4) follows from estimates (2.10) and (2,12) and the proof of the theoremis completed. 0

The next proP{"ity of A, established below. is crucial, not only in proving theexistence of solutions to \1.2) but in subsequent studies of approximations.

THEOREM 2.2. The operafor A: ')1 -+ ')1' defined in (2.3), safisfies fhe followingnonlinear Garding-fype ineqllalifY: 'V II, V E Bµ (0) c,Y.

[A(u) - A(d. II - v] ~ aoaollll - l'II~2(O.T.HAIO))+ a liliu - dill'- a2(µ )llu - vll£'°lo).

where H~(O) = wk2 (0) and an>O. al >0, a2(µ »0.Proof We observe that, 'Vu, v E Y,

(2.14) [A(II) - A(v). II - v]~ [A ,(u) - A I(V), II - v ]-I[A2(1I) - A2(v). u - v]l.

From the inequality in Ii" [10).

(2.15 ) I 1,-2 I 1'-2 ) 21-'1 I'( x x -}' y, x - Y ~ x -}' ,

(2.16)

and (2.6), it follows that

[A d II) - A dv), II - v] ~ an(l + c2(2, n) mes (0)2/") -111111 - villi 'en. T: nt'..2m),

+ kn21-P(l + cl'( p, n) mes (0)1'1" )-111111 - 1'1II1'.

On the other hand, according to (2.12) and Young's inequality for II, l: E Bµ(O)c l' and

any b >0.

GAl.ERKIS APPROXI~1A T10l'iS 921

(2.17)bp p' ",+,-111'YJµ ,

I[A:lu)-A2(v),li-L']I~-llIu-L.·IIII'+ - ".,' lIu-dr-col'p p

Therefore. by introducing (2.16) and /2.17) into (2.14) and choosing b smaIlenough. the desired result (2.13) is obtained. 0

THEOREM 2.3. Foran)' data IE ,,/1" and liuE L =(0). fhere exiSfS af leaSf olle SOllifiollII E "IV fo problem (2.2),

Proof. Theorems 2.1 and 2.2 and Aubin's compactness theorem [1] confirm thatconditions in [8] are satisfied. Therefore. A is coercive and 'W-pseudomonotone fromV -+ r and, by virtue of Lions [6, Chapt. 3. Thm. 1.2), the assertion of the theoremfoIlows. 0 .

Remark 2.1. From the proofs of Theorems 2.1 and 2.2. it is apparent that theoperator A of (2.3) regarded as a map from\' into \,'. is bounded. coercive and 10caIlyLipschitz continuous. and satisfies the Garding-type inequality

(2.181(A(u) - A (v). II - v) ~ aOoll"u - r1li/~/Oi + a diu - L.'III'

-a21pl\lrt - L.:lIr.,ol 'VII. L.' E Bp(O)c V.

According to Oden [8]. A: \. -+ V' is necessarily V-pseudomonotone. Hence. from thetheory of pseudomonotone eIliptic equations let. [6]). A is surjective from V -+ V'; i.e ..there exists at least one solution in V to the stationary problem

(2.]9) A{II)=f, f givcn in V'.

Thc cvolution problem (2,2) possesses at least one equilibrium state for each Ie V'.

3. Sufficient conditions for uniqueness. We now proceed to determine sufficientconditions for uniqucness of solutions to the pseudomonotone diffusion problem (2.2).

In the case of monotone parabolic problems. "monotonicity" ~ "uniqueness" andthis foIlows from the earatheodory type differentia] inequality dllt(t)-V(f)12/dt~0,a.e. f E [0. T].lu (0) - L'!0112 = O. the unique solution of which is III (f) - L.'(t 112 = O. It andv being solutions of the problem. This suggests that in the non monotone case withGarding-type inequalities. the possibility of establishing a differential inequality of theform

(3.1 )

or equivalently,

(3.2)

:rlu(t)-v(t)I' ~alu(t)-v(t)I' a.e. fE[O. T].

lu(O) - v(O)I' = o.

IU(T)-v{T>!';2a rT!u(t)-v(f>!'df 'VTE[O, T)Ju

for some a E Rand 2 ~ s < 00, would be sufficient for concluding uniqueness. Indeed.from Olech and Opial [9. Thm. 3]. lu(t) - dtll' = o. is the unique solution to (3.] l. Weshow that in certain particular cases and. in gencral. for sufficiently smooth solutions.problem 12.2) falls into this class.

922 G. ALDUNCIN AND J. T. aDEN

(3.5)

THEOREM 3.1. Lef U E lV be a SOIl/fiOIl of problem (2.2). Then u is uniqlle ill fhefollowing fhree cases:

i) r = ° and q = 1;(3.3) ji) r=O and n <p;

jii) ao> 0 and u E L cc'(o. T; wAxtm).1Proof Assume that l/ = 1I(f; f, uu) and L' = L'(t; f, uu) are two solutions of problem

(2.2) and define T/ = u - I). It is apparent from 12.16) that

(3.4) (A1(u(t»-A1(l,:(t). T/(f)~auaull1'/(t)IIi,,~(nl+a&T/(nll" for a.e. tE[O, T],

where au>O and al >0. Thus, from the difference of the equations satisfied by II andt', we obtain the integral inequality

!IT/l T )12 + aoau r 111'/(f)lIi/Acnl dt -:a.\ < =- Ir (Az( II (t)) - A 21 L' (I)). 1'/ltI) dfl·au>O VrE[O, T].

We now estimate the right-hand l)ide via the formula (2.111 with H' = 1'/.i) r = 0 and q = 1. In this case we have the estimate

(3.6) .\, ~ c« J ~11'/(tlI2 dt 'VT E [0, T],I)

which combined with /3.5) gives the integral inequality 13.21 with (t = 2cq and .5 = 2.eonsequenlly, 1'/= 0,

ii) r = 0 and n < p. From the Sobolc\' embedding theorem. WI~'''( 0) is continuouslyembedded in Culm = {to E Clm: L' bounded in n} whenever II <po Then

/3.7)

Let µ bc chosen such that II. l' E B ...(0) c: '}'. Then. from (2.1]), with II' = 1] andusing 13.7), we obtain, 'VTE[O, T).

2p2 -:a s = < p.

p+l-q

(3.8)

1 <

A< ~J f cqll~(tlIlF:!11IT/(t)12 dfdtJn II

, ~Cqµq-l(rl1](t)I'dffi',

Introducing this estimate into (3.5) produces the integral inequality 13.2) with

2 q-l I/' 2pa=( Cqµ ) - and 2~s= <po(p+1-q)

Therefore, T/ = O.iii) ao>O and l/EL""W, T: W\\·cc(O)). Let µ>O be such that lI.l·eB ...(O)c:

L ""(0, T; wkoc (0). Then, from (2.11) with II' = 1]. we obtain VT E [0. T],

(3.9) AT ~ µ""'-1 lcqc(2. n) mes (n )11" + c,) f T 111'/(t 1II/l,',l£l,I1] (t JI df.u

I In this case, the question of existencl' "ppear' 10 he open,

GALERKIN APPROXIMATIONS 923

(4.2)

Hence, since by hypothesis Go>O. we can apply Young's inequality with constant b,,-e.g .. b = vanGu. to obtain the upper bound for (3.9)

aoGn f ~ 2 a f ~I .,2 Jo

ll11ltIIlH,I"lIIdc+2' Ju

11(()I-df.

where a = a( l/b~) > o. eombining thcse results with (3.5) gives (3.2) with s = 2, and11 = O. :I

4. Galerkin approximations. In this section, we study Galcrkin approximationsof the model problem 12.2) which arc based on an elliptic regularization of 12.2)obtained as suggested by Lions [6]. We will establish some results on thc scrongconvcrgence of such approximations.

For the model problem (2.2), we introduce the elliptic regularization

14.1) Flli,. i')~-(II,. i')1(+llIfITl. rITIl+[AIII,), v]=[J. r]+(lIo, 0(0)). 'VrE:l/.I.

where A is the operator defined in \2.3). Following standard techniques discussed indetail by Lions [5] and [6], it can be shown that for every f > O. there exists at leastone solution II, e v7J. to 14.1) and. that, in the sense of 1" C 2'(£t«O. T)), W-I.I'·(O)), u,satisfies the distributional equation

-Fii,+li, +A(II,)=f in 'V',

-Fli,IO)+ II, (0) = lin in L 21m.

li,(Tl=O inL~(Ol.

which is equivalent to 14.1), Moreover, for any sequence {uf},>oc 1J1 of solutions.there exists a subsequence. also denoted {UE},>II' such that. as F -+0+. u, convergesweakly to a solution II of 12.21 in the sense:

II, -II weakly in r.nil, all

weakly in ''V'.---ill af

(Ill f

weakly in It.f--()ilf

(4.3)AlliE )-A(II) weakly in "V',

11.(0)-11(0) weakly in Je.

u..(Tl-u(T) weakly in 'jf

To construct Galerkin approximations of (4.1), we introduce a family of subspaces{qJ,,}o<h:li I of CJf.L such that: i) !ilLh is finite-dimensional with basis functions{<p I, <P2, ... , <Pm.J. with dimension nth -+00, as It -+0·; and ii) U .. JUh is dense in <1f1.. AGalerkin approximation of (4.1) involves seeking a function U~ E 61lh such that

(4.4)£(U~'. 4>kh -(U:~. cbk)l(+(U~'(n, c/>k(n)+[A(U:'I. c/>k]

= (f. <pd + lilli, C/>dO)), k = 1. 2 ..... nth·

The solvability in1fl" of (4.4) is assured by the u7L-pseudomonotonicity andcoercivity of the operator .w.: I1fI -+ Jij' in (4.1). Similarly. as for (4.1 ), if {U~ h,<.:J.;> I is asequence of Galcrkin approximate solutions, it can'be shown that there exists a function

924 G. ALDUNCJN AND J. T. ODEN

U, and a subsequence, also denoted {U~}O<I'=;t. such that, as h -+O~,

(4.5)

U;~li,A(U~)~A(u,)

U~(O)~u,(O)

U:(Tl~u,(T)

weakly in r,weakly in L 2(0),

weakly in r',weakly in L2(0),

weakly in L ~(O).

(4.6)

We will now demonstrate that for our model problem (2,2) much stronger resultscan be obtained,

THEOREM 4.1. Lef {u, },>o c: ul1 be a weakl}' convergenf subsequence of so/wions fOproblem (4.1) and with its weak Iimif U E Wa Solufion of (2,2). Then. as F -+ o~.

U, -+ u sfrongl}' in 'V,

J~li, -+ 0 sfrongly in L 2( 0),

u,(Ol-+ Uo strongly in L 2(0),

U, ( T 1-+ II (Tl sfrongl}' in L 2(0),

Proof. We regard equation (2.21 as holding on :5lJ and subtract (4.1) from it. Thefollowing orthogonality condition is obtained:

(4.7) -(eu .. v):I'( + (uo - u. (0). L' (0)) + [Ii - Il" v] + [A (II) - A (u,), v] = 0 'Vv E Il/L.

According to (4.3), there is a µ > 0 independent of F. such that II" II E B,. (0) c: -r. Hence,using formula (1.4) and the Garding-type inequality in (2,13). we see that

Next, combining these two results, we conclude that

1 2 1 '2111u -II. (0)1 + 21u (n - II. (T)I- + £l'drill- u, IIII'(4.9) ~ £l'2(µ )llu - II, lIf'p(QI + (Uo - II, (0), Un- v(O))

+ [u - Ii,. II - L:] + [A (II) - A (II,), II - v] + (eu" i')x -IJ~li, I~ 'VI: E GU.

Due to the compact embedding of 'W in LI'(O) (d. [1]) and the weak convergenceresult (4.3), (4,6) follows. 0

THEOREM 4.2. Lef {U~ E OlJ"}Il<.I'::i1 be a subsequence of Galerkin approximtltt.'solwions defined by (4.4), converging weakly. in the sense of (4.5). to a solwion II, E au ofproblem (4.1). Then, for fixed F > 0, as h -+ 0+

14.10)

u~-'u,

[j~ -+ Ii,

U~ (0) -+ II, (0)

U~ (Tl-+ II, IT)

sfrongly in r,sfrongly in e(O),

strongly in L 2(0),

sfronKly in L 2( 0).

GALERKIN APPROXI~fA nONS 925

(4.111

(4.13)

Proof. We follow similar arguments to those given previously, Restricting 14.1) toOUItand subtracting (4.4) from it, we obtain the orthogonality condition

elli, - U;. ",')K-(U. - U~I. W)K+lu,ITl- U:(T). WIT))

+[Alu,)-A(U~l. W]=O 'VH'EllJ,.

Now. from (4.5) and (4.3). there is a /L >0, independent of h. such that U~, u, E

B..(O)c'l'. Then. by virtue of 11.4) and (2.13). it follows that

e114,- c..i:12-(u, - U;, Ii. - c..i: h·+ju,( Tl- U;(nI2+[A(u. )-A( U~ l. u, - U:l

(4.12) ~ eili. - V; 12+ tlu.(O) - U~ (0)12 + tlu,( n - U~ (n12

+aollu. - U: II~ '(0.T;H~lllll + a dllu, - U: IIIP - a2(/L )lIu, - U: 11[",0"Therefore, combining (4.11) and (4.12),

, . h' 11 It 12 11 h 12clti. - U, Ix+ 2 u.(o)- U, (0) +2 u,(Tl- U, I Tl

+aollll, - U:'lIi'to.T;H.\IOIl+adllr" - U~'IIII'<: )1\ ' 7"111" , • U·,·· ,i,)=a2(/L U,-V,IL"\Ot+el,u,- ,.U,-n)(

It.· h-(u, - Un U, - Whv+(u,( T)- U, (Tl. u,( T)- W( TlI

+[Alu,)-AlU;l.u.-W] 'VWE5f1".

But 1U is compactly emhedded in e(Ql (1) and U: converges weakly to u, inthe sense of (4.5). Hence, the right side of (4.13) -+ 0 as h -+ 0'. and this proves thetheorem. C

We next give an error estimate for the Galerkin approximations of the regularizedelliptic problem 14.1).

THEOREM 4.3. For fixed e >O. let II, E 5f1be a soillfion of problem 14.11. which isfhe sfronR limif (in the sense of (4.10)) of the subsequence of Galerkin approximafesolutions {U~ E JUh}O<I, ~ 1 defined by (4.4). Then fhe following approximafioll erroreSfimafe holds 'VW E 5f1":

I ,,' 1 It'2Ct!U,(O)- U, (OW +2Iu.(T)- U, (TW

+aoaullu, - U;lIi'to,T:H,\HIJI +cilliu. - U~III/ + illi, - c..i; I~14.14) " . ,~a211u, - U.Ilr.".o,-C2Iur(O)- W(O)I-

+ C31u, - WI;"+ tillu, - WillI" -l- C114, - WI~.where Ci• i = I. ... ,4. (ro. a I = ada d. a 2 = a2( T. µ).i = i (I' ) and C = CI C( T. /L)) aresfricfly posifit.'e COllSfanfS.Here C( T. µ) is fhe local UpschifZ continuifY COll.Hanfof(2.4),

Proof. The estimate (4,14) follows directly from (4.13) upon applying formula(1.4), the local Lipschitz continuity of A, (2.4) and Young's inequality. 0

5. Faedo-Galerkln approximations. Wc are concerned here with Faedo-Galerkin approximations of the modcl pseudomonotone diffusion problem (2.21. Wenote that this type of approximation process is not neccssarily well-dcfined fornon monotone parabolic problems: the corrcsponding weak convergence is a condi-tional propcrty. We shall show that Faedo-Galerkin approximate solutions to prohlem12.21 exist and arc uniqlle. and we shall determine sufficicnt conditions for weak andstrong con vcrgence.

926 G. ALOUNCIN AND J, T. ODEN

(5.1 )

Let {Vh }O<II:i 1 be a family of finite-dimensional subspaces approximating the spaceV(= W~·I'(O)) in the following sense: (it {lh w:.· .. ,l/JmJ denotes a basis for Vh• withdimension m" -+ 00 as h -+ 0·; (ii) U I, Vh is dense in V. A Faedo-Galerkin approximationin Vh of problem (2.21 is defined as an absolutely continuous function Uh eCA ([0, T]: V,.). which is a solution of the system

(Cj"(t). wk)+(AIU"U)), I/1k)= (f(t), Wk). k = 1,2.···. mho

V"WI= U~,

for a.e. Ie [0, TJ and where V~ -+ Ul) strongly in L 1(0) as h -+ 0+. We observe that ifUh is solution of (5.11. then its time derivative (;" belongs to LI'·(O. T; V,,).

We next establish the solvability of problem (5.1).THEORE~1 5.1. For each h E (0. 1]. fhe Faedo-Galerkin approximafion problem

(5.1) possesses a unique sollllion Vi, E CAIrO. TJ; V,,) confinuollS Wifh respecl fO u~.Proof. The local existence of solutions to (5.1) in CA ([0, f,,]; Vh), I" > O. is implied

by the pseudomonotonicity property of A (cf. Remark 2.1). Indeed, fEY', and A isnecessarily bounded and demicontinuous from V .....V' and these are sufficient condi-tions for the vector field F(t. U) = «f(t), I/1k) - (A (U (tl), I/1k» from D = [0, T] x IRmh -+

(Rm" to satisfy the earatheodory conditions in D. Here U e IRm~ denotes the coordinatevector of U E V" with respect to the reciprocal basis of V,,,

The uniqueness and continuous dependence on the initial data of local solutionsto problem (5.1) follows from the condition [3J that for each compact set \I' c D, thereis a function g.. e L tW. Tl such that

(5.2) (t. U), (t, W)E w.

which is satisfied because A is locally Lipschitz continuous from V -+ V' (ct. Remark2.1).

It remains to be shown that the interval of existence [0, fh J = [0, T]. This is aconsequence of the coercivity of A from V .....V'. as follows from part (1) of the proofof Theorem 5.2. given below. CJ

We now proceed to analyze the convergence of the Faedo-Galerkin approxima-tion process.

THEOREM 5.2. From fhe sequence of Faedo-Galerkin approximafe sollllionsdefined uniquely br (5.1 ). fhere is a subsequence, also denoted {V" }n<h:i I. and fhere existfUllctions u e W' alld f{ E '7" sllch fhat. as It -+ 0".

u"- u weaklr in 'V.

Uh -u weakly* in L «'(0, T; L 2(0»,(5.3)

alld

A(U'I)-it'

Uh(n-u(T)

weakly in 7/",

weakly in L 2m),

(5.4) [ilU ]af,ll +[ff,vJ=[f.v] 'fIVEY, u(O)=un.

Moreover, fhe limit function u is a sollllion of problem (2.2) (i.e., ~ = A (u)) providedolle of fhe following condifions is safisjied:

(5.5) i) (;" e 'Ii', 0< h ~ 1, and {Ill Ljhll,*}.t<":iI is bounded;

(5.6) ii) A: 'V -+ 'V' of (2.3) is f/'-pseudomonofone.

GALERKIS APPROXIMATlOSS 927

15.9)

(5.10)

(5.111

Proof. We follow the usual pscudonomontone method which consists of: 11 findinga priori bounds: 2, passage to the limit: and 3) the pseudomonotonicity argument.

1) From the proof of the coercivity property of A, (2.8), it is apparent that A j!'.

also coercive from V -+ \l'. Then. by standard arguments. it follows that the sequence{V"}O<":i1 turns out to be bounded in Y and in L oc(O, T; L~(Ol).

2) With the previous result and the bounded ness of A from J . -+ '1" given by 12.5).the validity of (5.31 follows via weak compactness arguments and, then. upon thepassage to the limit in equation 15.1), (5.4) is easily concluded (d. [6. ehapt. 2]).

3) It remains to be shown that. if either /5.5) or (5.6) holds. then

(5.7) [~, t'] = [A(II). r]. 'VL' E Y.

From (5.1), (5.3) and (5.4). we see that

lim {CU"~, V"]+[A(V"), VI,]} = lim (f. Vi,] = [t. u]"10 "Ill

= [Ii. II] +[~:,u]

=lim{[li. V"]+[A(V"1. u])."10

Therefore,

(5.8) lim [A( Vi,). V" - u] = -lim [(/' -Ii, V"] = -! lim IV"( TJ -I/(Tlj: ~O.1,10 Ido "10 ...

Now, by the usual arguments [6]. 15.7) follows from (5.8) and the first statement of(5.3) when assuming either 15.5) and using the W-pseudomonotonicity property ofA: Y -+ 'V', or (5.6). This completes the proof of the theorem. C

We next establish that condition 15.5) is also sufficient for the strong convergenceof the approximation process.

Theorem 5.3. SlIppose fhe condifion (5.5) holds with bOl/lld µ' >n. Theil thesubsequence {V"}I1<,,:li I of Faedo-Galerkin approximafe solurioflS conL'erginJ{weakly toa solwion u E W' of problem (2.2). in flte senseof Theorem 5.2, is sllch fha f. as h -+ 0',

V" -+ U sfrongl}' in L ""(0, T: L 2(0)),

V', I . ""-+ II strong}' 111 Y.

In facf. the following approximafion error eSfimafeshold 't/Z E LPlO. T: V,: I:

I, ". I, 'rlu(r)-V (r)I~luo-Vol+KJ(T,µ)III1-U IIrl'l'O I

+K2(T,µ,µ')lIIu-ZIIII/2 'V7E[O, T];

11111 - V" III ~ K Jillo - V~ 12/P + K4( T. µ. )1I1l - VI, IIl~\'O-II)

+ Ks(T, µ, µ')lIIu -ZIIII/I'.

where µ >0 is a bOllnd for u and {V"}lI<J':li1 in ~I'.

Proof, By using formula (1.4) and the GArding-type inequality (2,13) In

LI'(O. r: W~·I'IO)). r E [0. T], it follows that

J.' (Ii(tl- U"U)+A(u(t)) -A(U"(tI). 1I(t) - U"U) dfII

(5.12) ~ }llIl r) - U"l 7)12 - ~II/o- V~;12+<t If' 1111(t) - U"It)W' cit"

T 1"11'

- <r2(µ' ) (1. 11/1 (n - V'' ulIIf."llll df)

928 G. ALDUNCIN AND J. T. ODEN

and, from equations (2.2) and (5.1). the following orthogonality condition holds:

(5,13) f'(li(n-V"(t)+A(II(t))-A(Uh(t)),Z(r)}dt=O VZEL"((), T: V,,).u

Hence, introducing (5.131 into (5.12) and using the local Lipschitz continuity property12.4), we obtain

!III tTl - U" (T lf + Il' 1fT 1111 ( t) - u" (nilI' d r(I

(5.14) ~:d'lIo - U~12 +Ct2(µ )llu - u"lIr-tOI + {C(µ )lIIu - U"III+ lliu - V"III.11ll1i - ZIIIVTE[O. n. 'VZeLl'lO. T: V).

Therefore. the approximation error estimates (5.10) and (5.11) are implied by (5.14).Note that the strong convergence of U" -+ Ii in LP

( 0) is a consequence of the firststatement of 15.3). assumption (5.5) and the compact embedding of -11"into [1'(01. 0

The pOfe1lfial case. As a final result. we shaH establish that if the bounded, coercivetocaHy Lipschitz continuous, Garding-type operator A of (2.3). is potential in thefollowing ser.se:

em. A is the gradient of some Gateaux differentiable functional J: V -.iR, forwhich there i's a constant .y >0 such that

(5.15 )

then. for data

(5,16)

(5.17)

l/. lio)eL~(O)x V.

U~ -+ 110 strongly in V.

(5.1H)

(5.19)

the Faedo-Galerkin sequence of approximations defined uniquely by (5.1) is sllch that

{U"}O<":i1 is bounded in L rW. T: V),

{U"}O<":il is bounded in L2(0).

Since e(O)'-f> '1". the second statement of (5.18) is stronger than (5.51 and, con·sequently. the results of Theorems 5.2 and 5.3 are true in this potential case.

We now prove this result and establish the corresponding regularity of limitfunctions.

THEOREM 5.4. Let the operator A of (2.3) safisfy condition eIIl and considerproblems (2.2) and (5.1) with dafa (5.16), (5.17). Then the Faedo-Galerkin seqllenceof approximate solutions {U"}O<h:i I is bounded in file sense of (5.18). Fllrfhermore. fhert'is a subsequence of approximations, also denoted {U"}O<h:iI' converginR strongly to asolution IIe 'IV of problem (2.2) in fhe sense of (5.9), such rhat, as h -+ 0·.

U" ~ II weakly* ill L 00(0, T; V),

U" ~ Li weakl}' in L 2(0).

Pmof. Lef (U"}O<J':i1 be the Faedo-Galerkin scquence dcfined uniquely by (5.1),(5.16),15.17), which approximates problem 12.2) with data (5.16), ~nd suppose alsothat cOlldition em holds. Then, by replacing I/Ik by V" in equation (5.1). integratingwith respect to time from 0 to TE rO. Tl and. then. ohserving that dJ1UI/(t))/df =(A(U"(r)). V/'(r)) and that (f(r), VI/(t»);ai!lf(t)12+~lu"(t)12 for a.e. f E (0, n. we

GALERKIf': APPROXIMATIONS

obtain

929

(5.20) ! f. Till' (tl12 dl + iillVh Ir)1I" ~]( V::) +! J. r Ifu )I~dr(I 0

'VrE[O. T).

But. from the boundedness of A as a map from V -+ V' lcf. remark 2.1 I.I 1

](v:;)=]o+J. (AlsV:;), V:;)ds~]n+J. IIA(sV~)II*dsIlV::II~const.II n

Therefore. (5.181 is true.Next observe that from Theorem 5,3. there is a subsequence of approxima-

tions {Vh}O<h:!il that converges strongly to a solution Ii of problem (2.2) in'Vn L colO. T: L2(0)). Hence. uh

~ u weakly in Y(~ L 1(0, T: Vl densely) and thistogether with the first statement of (5.18) is equivalent to the first statement of (5.19).Also {Vh}O<h:!il is bounded in OZL(~ Y densely) and this with Vh~U weakly in Y isnecessary and sufficient for Vh

~ u weakly in OZL (d. [11. * V. Ill. Then the secondstatement of (5.19) necessarily holds and this completes the proof of the theorem. "

Conclusions. For the nonlinear evolut'ion problems considered here. we haveshown that the existence conditions of coercivity and 1V-pseudomonotonicity ofA: V-+ V'. are satisfied and that. under conditions (3.3) uniqueness of solutions isguaranteed. The elliptic regularization ideas discllSsed in ~ 4 provide a generalframework for Galerkin approximations of coercive lV-pseudomonotone problems,We have established criteria for the existence and weak convergence of such approxi-mations, as well as strong convergence whenever a nonlinear Garding-type inequalityof the form

[Alv)- A( w), v - IV] ~ atilt.' - wlh' - H(µ..lIv - wlluco.T.x ,) 'Vr. II' E B" (0) c Y.

holds. Here a 1> O. and X is a Banach space continuously embedded in H and in whichv is compactly embedded. Also, if in addition, A: V -+ Y' is locally Lipschitz continuous.we have shown that error estimates for Galerkin approximations can be derived.

The Faedo-Galerkin method was considered as an alternative method for con-structing approximate solutions. In these cases, coercivity. boundedness and demicon-tinuity of A from V -+ V' are sufficient conditions for existence. and local Lipschitzcontinuity from V -+ V' is a sufficient condition for uniqueness. As we have seen. theconvergence of this method is a conditional property in the case that A is nonmonOlone:the Faedo-Galerkin method is weakly convergent if: (i) the sequence of time deriva-tives of the approximate solutions is bounded in Y': or (ii) if A: 'V-+ Y' is Y-pseudomonotone. The convergence of the method is strong if: (iii) condition (i) holdsand A is locally Lipschitz continuous and satisfies a nonlinear G~rding inequality ofthe type given above. Furthermore, in the case in which condition (iii) is satisfied. errorestimates are derivable which are compatible with the interpolation theory of finite-elements in Sobolev spaces [7], [2].

This establishes condition (i) as a fundamental convergence condition for theFaedo-Galerkin method when applied to coercive ~V-pselldomonotone parabolicproblems. In particular, we have shown that this condition is satisfied whenever A is.in addition: continuous and potential from V -+ V'. its potential is coercive. and thedata (f. Uo)EJi'x V. In this potential case, the convergence condition (i) holds in'J{ ~ -V': furthermore. the approximate solutions form a sequence bounded in

930 G. ALDUNCIN AND J. T. ODE~

L""(O, T; V) ..... ~-, and the regularity in time resu It .. (u, iJII/ cJr ) e L x (0, T; V) x J{"

holds for the exact solutions of the problem.

REFERE~CES

[I] J. P. AUBIN. Un /heoreme de cOl1lpacire, C.R. Acad. Sci. Paris, 256 119631. pp. 5042-5044.[2] P. G. CIARLET, The Fini/e Elel1lent Ml'IJlod for Ellip/ic Problel1ls. North-Holland. Amsterdam. 1978.[3] J. K. HALE. OrdillllT)' D((ft'Tt'I//ial EI/II<l/iollJ,Wile~-lnlerscience. New York. 1969.[4] J. L. LIONS. Sttr leSl'spaCt's d'illlerpola/iall ; dl/ali/e. Math. Scand .. 9 11961 I. pp. 147-177.[5] -. SI/r cer/ail/t'I eql/a/iolls pmabaliqlles 11011 lillta ire,!, Bull. Soc, Math, France. 93 119651.

pp.155-175.[6] -. QI/I'lqlles Merhadt's de Rt'soill/ioll dl'S Problemt's tll/X Limi/es NOli Lilltaires. Dunod, Paris.

1969,[7] J. T. ODE" AND J. :-:. REDDY. All [lltrodllction /0 thl' .\farhemarical 17leOr}'of Fini" Eleltll'lITs.Wiley-

Inlerscience. New York. 1976,[8] J. T. ODEN. E.ris/enct' theorems for a class of problel1ls ill nOlllinear elasticity. J. Math. Anal. Appl.. 69

119791. pp, 51-lB.[9) C. OLECH AND Z. OPIAL. Sltr WIt' inega lite diffhl'tltielle, Ann. Polon. Math .. 7 ( 19601. pp. 247-254.

[10) J. T. ODEN. C. T. REDDY AND N. KI KUeHl. Ol/alira/h'" t1fralrris and finite elem.·/IT approximation ofa class of IIonmon%ne I/olllil/ear 'Dirichle/ problem,l. T1COM Rep. 78-15, The University ofTexas at Austin, 1978.

[11] K. YOSlDA. Functional Anah·sis. 4th Ed .. Springer-Verlag. N~w York. Heidelberg. Berlin. 1974.