Existence, Uniqueness, and Numerical Analysis of Solutions of a Quasilinear Parabolic Problem Author(s): Dongming Wei Source: SIAM Journal on Numerical Analysis, Vol. 29, No. 2 (Apr., 1992), pp. 484-497 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2158136 Accessed: 23-11-2016 05:41 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/2158136?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Journal on Numerical Analysis This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms SIAM J. NUMER. ANAL. ? 1992 Society for Industrial and Applied Mathematics Vol. 29, No. 2, pp. 484-497, April 1992 010 EXISTENCE, UNIQUENESS, AND NUMERICAL ANALYSIS OF SOLUTIONS OF A QUASILINEAR PARABOLIC PROBLEM* DONGMING WElt Abstract. A quasilinear parabolic problem is studied. By using the method of lines, the existence and uniqueness of a solution to the initial boundary value problem with sufficiently smooth initial conditions are shown. Also given are L2 error estimates for the error between the extended fully discrete finite element solutions and the exact solution. Key words. method of lines, finite element method, L2 estimates, quasilinear parabolic problem AMS(MOS) subject classifications. 65N30, 35J65 1. Introduction. In this work, we show that, by using the method of lines, the quasilinear parabolic problem governed by the p-harmonic operator has a unique weak solution which is more "classical" than the weak solution obtained by applying the theory of Kacur [4], in the sense that it satisfies the equation pointwise with respect to time. Therefore, in finding numerical solutions to this problem, integration can be carried out only on the spatial domain. In the formulation of this problem integration over the time interval is not needed while it was needed in the formulation used in [4]. With this formulation, L2 error estimates for the error between the true solution and its fully discrete approximations are obtained. In [7] and [1O]-[12], the method of lines is extensively used. 2. An existence and uniqueness result. Throughout this paper, we shall assume that fl is a bounded convex domain in Rn with smooth boundary fl, and p _ 2. We also use u ( t) or simply u to denote function u (x, t) which is defined on fQ x [0, T], T > 0. We use the following notation u=[fuPcx]'~, 2=[ u2cix 1/2 IU 1 =[ IV Ul P dx ] , 11 U 112 I U 12 dx] II- 112 is the usual L2(fQ) norm and || the seminorm for W` P(f) which is a norm for Let A: W" P(f) -* ( Wl P(f))* be the operator defined by (Au, v) = IVuIP-2(Vu, Vv) dx for v E WI P(f). For definitions of Sobolev spaces Wl`(fl), Wl`(fl), and (W1'P(f1))*, see [2], [5]. We quote the following lemma from [3]. LEMMA 1. There exist constants a > 0 and 3 > 0, such that, for p ? 2, a IIu - vlP = (Au -Av, u - v) and IlAu - Av II* -< 13(11 u 11 +1 Vl v ly-2 _ 11 u _ v 11 for any u, v E W"P P(Q). Note. For p _ 2, L2(fQ) D WI"p(f). In following (*, *) is understood as the usual inner product in L2(fQ) and ( as the duality for a pair in W `(fl) x ( W1 P(f1))*. * Received by the editors January 16, 1989; accepted for publication (in revised form) February 13, 1991. t Department of Mathematics, University of New Orleans, Lakefront, New Orleans, Louisiana 70148. 484 This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms A QUASILINEAR PROBLEM 485 LEMMA 2. For any g c WP '(Q), the problem (Au, v) = (g, v) for any v E WOP(fQ), u Ian = UolaO has a unique solution u c W1P (fQ), where uo E WP 0 (Q). Proof. Since fl is a bounded set, we have ( W1,p(f))*D Wl.q(fD) =,p(D ) where q = p/(p - 1), and thus g c ( Wl.P(f))*. And by Lemma 1, A is a strictly monotone operator. Therefore A satisfies all the conditions in Theorem 29.5 [2, pp. 242-243]. By the conclusion of this theorem, the problem has a unique solution. Consider the following nonlinear evolution problem (1) du +Au =f; x Qfli t E(0, T], dt (2) u(x, t) = 0f(x), x c ofk, t E (0, T], (3) u(x, 0) = u0(x), x E f, where u0 c W'P (fQ), u0j,n = b and f: [0, T] -* L2(fQ) is Lipschitz continuous, i.e., there exists a positive constant L such that |If(t) -f(t') 112_ LIt - t'I for any t, t' [0, T]. Note. Here we only consider fixed boundary conditions since the method of lines does not apply to this problem with time-dependent boundary conditions. This is clear since (8) requires u(ti)-u(ti-1) Wo'P(fl). DEFINITION 1. Let u(x, t): [0, T] -* L 2(f). If there exists a function g(x, t) such that lim u(t + At) - u(t) _ g(t) =0, At-0 A&t 2 we then say that u is differentiable at t, and g(x, t) is called the derivative of u(x, t) at t, which is denoted by du(x, t)/dt. DEFINITION 2. We say that u is a solution of (1)-(3) if u(x, t) c W1'P(fQ) for all t (O, T], (4) Kdtu v +(Au, v)=(f v), (5) (u(0), v) = (u0, v) for any v c Wo P(Q) and (6) u(x, t) = f(x), x E afd, t E (0, T], where du(x, t)/dt is the derivative in the sense of Definition 1, u0E W'p(Q), uolIn = f(x). THEOREM 1. Suppose that uo E W P(fQ) and V _ (IVu0Ip-2Vu0) E L2(fQ), then problem (1)-(3) has a unique solution u in the sense of Definition 2. Furthermore, u C[O, T; W1'P(fl)] and du/dt c C[O, T; Wh'P(fl)]. Let {ti}i=o0n be uniform partition of [0, T], At = T/n, and ti = iAt. Consider the following recursive nonlinear elliptic problems. Given uj_j, find ui such that (7) (u 1Ui v) + (Aui, v)=(f,v), (8) ui=uui1, on df forany ve Wo'P(f), where ui = u(x, ti), f =f(x, ti), i = 1, n. This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms 486 DONGMING WEI Lemma 2 above assures that for each such partition {tj}j=0,,, (7), (8) can generate a unique sequence {u1}ui n in w' P(Q). To prove Theorem 1, we first establish several lemmas, namely Lemmas 3-7, under the hypothesis of the theorem, i.e., V _ (jVuOjp-2Vu0) c L2(fQ). In the following C(uo,f) denotes a generic constant depending only on u0 and f LEMMA 3. For the above sequence {uili=O,n, there exists a constant C(uo, f) such that (9) ||A =| C(Uo,f)) At 2 Proof. In (7), let i = 1, v = (ul - uo)/At. We have U 2 1 K l u- uo (A ui-uo) ||' |+ Z(Auj -AuO,ujl-u"o)= (fl, 1Z )- (Auo, I ) At 2 At At At which implies that (10) 1 jt f2 flI211 uuAt 2 ( Auo,u) since, by Lemma 1, ( 1/A t)(AuI - Auo, uI - u0)o 0. Applying the divergence theorem to the second term in the right-hand side of (10) and using the fact that u1 - u0Ez Wo'(fQ), we have 2 u1-u0 ~~~~V (I'Vuop2,V uo)( u)d At 2 Alli211 At 2+ A u01t A dx -'(jlfl 12+ IV 1(VuoIP2Vuo)112) 9u'- At 2 and hence obtain A ut < (lfl 112 + liV (1V1o22Vuo) I2). Since, by letting v = ui - ui-1 for i ? 2 in (7), (- Ui1i K 9 U- UAti + (Aui, ui - uj-1) = (figU Ui-1)1 KUi-1 - Ui-2 U - ij + A i,u - i, =f-, iu- At we have (iAt ' i Ui-ui l+(Auj-Auj_ls uj-uj_l) K =u, u1 i-\?i-ui-lA + u(f--19ui-ui-1) At which implies, by Lemma 1 again, Ui -Ui 11 ui - Ui-112+a1ui -ui-l1P At 2 __ Ui-uI -Ui2 +I-11)~u-, 2 At 2 /-i112 jU _u _12 This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms A QUASILINEAR PROBLEM 487 And hence, by (12) and the Lipschitz continuity off, we have 11U, - Uj 1 C1 Ui-I - Ui-2l+ 1--l2 At 2 At 2 (13) -||Ui_l Ui_2|| +AtL. At 2 By (11 ) and (13), we finally have ||Ui-Ui||_ Ul-UO + TL At 2 At 2 (14) ' ( lflf112 + IIV (lVuoP12Vu)112) + TL. By (14) and the regularity hypothesis on uo, i.e., V _(iVuojP-2Vuo) c L2(fQ), we then have (9), with C(uo,f) = MaxO_t T IIf(t)JJ2+ I|V _ (jVuolp-2Vuo)II2+ TL. The proof is completed. As a consequence of Lemma 3, we have the following. COROLLARY 1. For the sequence {u}uj=o,n in Lemma 3, there exists a constant C(uo f) such that llUi 112 ' C(u0of), i = 1, n. LEMMA 4. There exists a u* c L2(fQ) for each i, such that (Aui, v) = (u, v) for any v E Wo'P(fQ) and J1 Aui 11 * = 1j U *11 2, where i = 0, n. Also 11 u* 112 C(u0o f ) for some constant C (uo, f). Proof By (7), we have, for i = 1, n, (Aug, v) = (Ui-ui, v) +(f, v) for any v EW P(f). By (9) we know that Aui is a bounded linear operator on Wo'P(fl) with respect to the norm; in fact, JlAuj1 *= 1j((uj-ui_l)/At)+fl2 _C(uo,f). Also Wo'P(Q) is a subspace of L2(fQ); in fact, this is a compact imbedding. Therefore by the Hahn-Banach theorem [8, p. 111], Aui can be extended to a bounded linear operator Fi on L2(fQ) so that jF 1 2* = IjAujll*. Hence, there exists a u e L2(fQ) with II Fi 11 * = 11 u* 112, and Fi(v) = (u*, v) for any v E L2(fQ). In particular, (Aui, v) = Fi(v) = (u*, v) for any v c W`P(Q), and IIu*112= lIAui|lI_ C(uo,f). COROLLARY 2. For the sequence {ui}i=O,n in Lemma 3, there exists a constant C(uo,f) such that lluill- C(uof) i=1, n. Proof By Lemma 1, Corollary 1, and Lemma 4, we have a11 ui - uoII P (Aui - Auo, ui - uo) = (u - u*, ui - uo) (15) _ (|i1 'I12+ 11 U*jj)jjUi - U0112= C(U0,) i = 1 n- By the convexity of | jP, we have lluilP = 2P-'(Ilui - uollP + Ill"IP), which, together with (15), gives the result of this lemma. This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms 488 DONGMING WEI Now, let {tj}=,,,n and {tk}k=0,n be two uniform partitions of [0, T], t-t ti?1-t un(t)= atAt i+, + a t t", t ti + , i=0,1, n t -tk +tk+1 -t Um(t) = Uk+1 + , tk < t?- tk+l, k=O, 1, m-1, Un(0) = U.(0) = U(0), ati = iT Atk = k n m Let Yn(t) =Ui+1 for ti < t ti+, i = O, 1, 2, n *, n-1 U(0) =UO, Yrn(t) =Uk+1 for tk < t?tk+l, k =O, 1, 2, * - *, m-1, um(O) =uo. Obviously, dt A t ,i ti < t-ti'+I dUm(t) Uk+l Uk dt - Atk tko dt / uniformly in [0, T]. For each t E [0, T], by Remark 1, {dun(t)/dt} is a uniformly bounded sequence, with respect to t, in the reflexive Banach space L2(Q) and hence has a subsequence which converges weakly to an element w(t) E L2(fl). Thus, we have, by (27), that (28) (w(t), v)+(Au(t), v)=(f v) for any ve W' PM). This w(t) is independent of the subsequence, since for fixed u and f (28) has only one solution. Since the weak limit of a uniformly bounded sequence is also uniformly bounded [2, p. 193], w E Lc(0, T; L2(fQ)). Therefore, again by the Hahn-Banach theorem, (28) can be extended to hold for any v E L2(fQ). Let t, t'E [0, T]. Using (28), we have (29) (w(t) - w(t'), v) = (Au(t) -Au(t'), v) + (f(t) -f(t'), v), which also holds for any v E L2(fl). Let v = u(t) - u(t'). By (29), Lemma 1, and the boundedness of w in L2(fQ) norm, we get a jju(t) - u(t')j1P < (Au(t) -Au(t'), u(t) - u(t')) (30) = (w(t) - w(t') -f(t) +f(t'), u(t) - u(t')) c(Uo,f)IIu(t) - u(t')jj2. This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms A QUASILINEAR PROBLEM 491 By Lemma 6 and (30), we get (31) lim 1 u (t) - u (t')l = . Thus, u E C[O, T; Wl"(fQ)]. We next show that we C(0, T; L2(fQ)). By Lemma 1, we have 11 Au (t) -Au (t') 11* --<<:8(1 u (t) 11 + llU(t,)II)p-2IIU(t) - u(t) |- Therefore, limt t' 11 Au (t) - Au (t') 11 * = 0, since u E C [O, T; W" P (f)] . Since w,fe L'(O, T; L2(fQ)), by the Hahn-Banach theorem and (29), there exist u*(t), u*(t') E L2(fQ) so that (w(t) - w(t'), v) = (u*(t) - u*(t'), v)+(f(t) -f(t'), v), for any v eL2(fl). And (32) lim 11 u*(t) - u*(t') 112 = IIAu(t) - Au(t') II* = 0. t --> t' Let v = w(t) - w(t') in (29). We get w(t)-(t'l2_ = I(Au (t) -Au (t'), w (t) - w (t')) I + I|(f(t) -flt'), w (t) - w(t')) -' ( 11 u*(t) - u*(t') 112+ Ilf(t) -f(t') 112) 11 w(t) - w(t') 112, which gives (33) IIw(t) - w(t')112? (IIu*(t) - u*(t')II2+ lIf(t) -f(t')112). Therefore, (33) and the continuity of f imply that we C(0, T; L2(fQ)). Let u*( t) = JO w(s) ds + uo. Using Fubini's theorem, we have (u.(t) - U*(t),vV = f| (-n_ w) V ds dx = f f ("d n-w) vdxds (34) =W v ds Jj dt WVU Jof [(dtn < - (Au, v)+(Jf v) ds. Thus, by (27), limn-O (un(t) - u*(t), v) = 0 for any v E Wo'P(fl), uniformly over [0, T]. We have u (t) = u*( t) =JO w(s) ds + uo, since the weak limit is unique. We now show that u is differentiable in the sense of Definition 1. In fact, without loss of generality, let At> 0. Then, we have u(t?+ t) -u(t) - ()2 it+At 2 ( i)(-w(t) 2 1 2 w(s) ds-w(t) f[1 ft+At (w(x, s) - w(x, t)) ds] dx f [ t+At Iw(x, s) - w(x, t)I ds] dx. By Jonsen's inequality [8, p. 63], we get u(t At) - u(t) 2 t+-t ds) -W(t) | (W(X, S) W(x, t))2 dS dX 1 r t+At (35) - ~~~~ J~~ ~~J (W(x, s) _ W(x, t0)2 dX)d - t+At 11 w(s) - w(t) 112ds =II w(6) - w(t) 112, where t _ t + At. This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms 492 DONGMING WEI Hence by (35), limAtO ||(U(t + At) - u(t)/At)-W(t)II2 = O, since wE C(O, T; L2(fQ)). We get du/dt = w. Finally, by (28) and Definition 1, we get (du v)+ (Au, v) =(f,v) for any v c W1'P (Q), in [O, T]. This completes the proof of the existence of a solution. For uniqueness, let us assume that u and u^ are two solutions to the problem. Then, (36) (dX v + (Au, v) = (f, v) for any v E Wo'P(fl), in [0, T], and (37) (dt, v + (Au, v) = (f, v) for any v E WoP(fQ), in [0, T]. Subtracting (37) from (36), we get (d -du,v)+(Au-Au,v) =O foranyvE W'P(fl). Let v = u - u. Then, we have Kd(u -1a)A _A) dt ) +(Au-Au,u-,u=O, i.e., dt (11 U- 2 + (Au-AAu, u-iu)=O. Since, by Lemma 1, (Au -Au, u we have d (|U _ All 2)-O. dt 2 IIu(t) - i(t)II is therefore a decreasing function in [0, T], and therefore 1I u(t) - A(t)12.< IIU(O) - A(O)11I2 = IIUo- Uol4 2 = 0, for all t in [0, T]. This completes the proof of Theorem 1. 3. L2 error estimates for the fully discrete scheme. Let Sh (fQ) be a conformal finite element space of W P(fl) as constructed in [1, (5.3.5), p. 313], and let Hh: W"4(f) -* Sh(fl) be defined by IlhU = Em=L1 l(u)N, [6, Vol. iv, pp. 63-64]. Hh is known as the finite element interpolation operator; {Nj}i=i,m are the global basis functions for Sh(fQ) and {li(u)}ij=,m correspond to the global degrees of freedom. A classical theorem on global interpolation error estimates in the finite element theory [1] leads immediately to the following. LEMMA 8. Suppose that {Th}h is a regular family of triangulation of fQ. We then have, for p ' 2, the following interpolation error estimate: IIu-HhUII _ ChIU12 foruE W2u (Q), where IU 2 is the L2 norm of the second derivatives of u, C is a constant independent of u, h is the maximum of the diameters of all the elements in {Th}h, and HhU is the finite element interpolation operator. This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms A QUASILINEAR PROBLEM 493 Remark 2. If Hh is the interpolation operator defined in [9, (2.12)], then we have I1U -FIhU IIP _ Ch 11 u for u e W' P(Q). Again for simplicity, let {tj}j=0,, be a uniform partition of [0, T] and At = T/n. Let {Ui}i=0,n be the sequence generated by (7), (8). For each i consider the following problem. Find Wi c Sh (f), such that (A Wi, V) = (Aui, V) for any VeSh (f) n W P (fi), i = 0, n, (38) 0 Wilan = HhUi Ian By Theorem 29.5 of [2], for each i, problem (38) has a unique solution. LEMMA 9. II Wi || C(uo,f) i =0, n. Proof. In (38), let V= Wi-HIhUo. Then (AWi, Wi-HhUo) = (Au, Wi -HhUO), i.e., {V WiP dx-j IVi?I-2(VW,VHhuo) dx = Vilp-2(Vui V Wi) dx- IVUilp-2(VUi VHhUO) dx. We hence get I vWiIP dx u I | P-IV WiI dx +j IVWiIp-'IVHhUoI dx + fIVuilp-IVIhuoI dx rr 1 (p-1)/p r 11/p c; J IVuiIP dx [LIVWiIP dx r . ~~~~(P-1)/p r(P-')/p] 1 /p + {Lf. IV WilP dx + IVuilP dx IVHIuoI" dx]. i.e., (39) || Wil || - P cIHhUoI|(I| Wi |P 1+ ui |I) + || ui||P li ijj| From (39) and Lemma 3, the conclusion of this lemma can be obtained. LEMMA 10. IIuI-hI/I _ C(u,f)(IIui-11hUi I) "(P ), i = 0, n. Proof By (38), we have (A Wi - Aui, V) = 0 for any Ve Sh(n)fn W!,P(n), i=o, n. In particular, (40) (AWi-Aui,Ilhui- Wi)=0, i= 0,n. By (40), we get This content downloaded from 178.91.253.59 on Wed, 23 Nov 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms 494 DONGMING WEI By Lemma 1 and (41), we get a 11 ui -Wi || P (Aui -A Wi, Ui I-IhUi) |lAui -A1WiI|*|ui -fIhUi |3 (II Ui I| + 1V Wi -I) || Ui 1VWil | Ui JhUi |,I which gives (42) a || ui- Wi 1 P1 c3 (IIuiI + |1|WilI)||21Iui HhUi 11 By Lemma 9 and (42) we get the result. Now, we consider the fully discrete scheme: Let U0 = W0, where W0 is defined by (38). Find Ui E Sh (f), such that (Ui A 'U_ V +(AU,, V)=(f, V) forany VeSh(fQ)fn Wi'p(), Ui 1 Cubics at the Vertices of an Underlying Triangulation [pp. 528 - 533] Exploiting Symmetry in Boundary Element Methods [pp. 534 - 552] Corners, Cusps, and Parametrizations: Variations on a Theorem of Epstein [pp. 553 - 565] Lattice Integration Rules of Maximal Rank Formed by Copying Rank 1 Rules [pp. 566 - 577] Error Bounds for Gaussian and Related Quadrature and Applications to R-Convex Functions [pp. 578 - 585] Optimal Nodes for Interpolatory Product Integration [pp. 586 - 600] On Grau's Method for Simultaneous Factorization of Polynomials [pp. 601 - 613] Back Matter [pp. 614 - 614]