A SHADOWING RESULT WITH APPLICATIONS TO FINITE ELEMENT APPROXIMATION OF REACTION-DIFFUSION EQUATIONS STIG LARSSON AND J.-M. SANZ-SERNA Abstract. A shadowing result is formulated in such a way that it applies

in the context of numerical approximations of semilinear parabolic problems. The qualitative behavior of temporally and spatially discrete nite element solutions of a reaction-diusion system near a hyperbolic equilibrium is then studied. It is shown that any continuous trajectory is approximated by an appropriate discrete trajectory, and vice versa, as long as they remain in a suf ciently small neighborhood of the equilibrium. Error bounds of optimal order in the L2 and H 1 norms hold uniformly over arbitrarily long time intervals.

1. Introduction The purpose of this article is to compare the dynamical system arising from a semilinear parabolic evolution problem with the dynamical systems that arise from its temporal and spatial discretizations. The long-time behavior of a dynamical system is governed by its invariant sets such as xed points, periodic orbits, attractors, etc. It is therefore important to investigate whether the discretized dynamical systems have the same kinds of invariant sets and whether their orbits have the same qualitative behavior near these sets. Our aim here is to do so for the special case of a hyperbolic xed point. The inspiration for this work came from an article of Beyn, [3], on multi-step approximations of systems of nonlinear ordinary dierential equations, u0 = f (u). Beyn showed that if the continuous problem has a hyperbolic xed point u, then there is a neighborhood O of u such that the following conclusion holds: for each initial value u0 2 O there is U0 2 O such that the approximate orbit U starting from U0 is close to the exact orbit u starting from u0 as long as the latter orbit stays in O. The error u ? U satis es an estimate, which is uniform with respect to u0 2 O and of optimal order of convergence. Note, in particular, that the error bound is thus uniform over arbitrarily long time intervals. The converse statement is also true: for each U0 2 O there is u0 2 O such that the corresponding orbits U and u are optimally close as long as they remain in O. We emphasize that U0 6= u0 Date : January 1996. 1991 Mathematics Subject Classi cation. 65M15, 65M60. Key words and phrases. Shadowing, semilinear parabolic problem, hyperbolic stationary point, nite element method, backward Euler, error estimate. Published in Math. Comp. 68 (1999), 55{72. The rst author was partly supported by the Swedish Research Council for Engineering Sciences (TFR). The second author was partly supported by \Direccion General de Investigacion Cient ca y Tecnica" under project PB92{254. 1

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STIG LARSSON AND J.-M. SANZ-SERNA

in general, because the initial value problem is typically unstable near a hyperbolic xed point. Beyn's result was extended to in nite dimensional spaces by Alouges and Debussche [1], who were thus able to cover pure time-discretization of semilinear parabolic equations. Another extension to semilinear parabolic problems was made by the present authors in [12], where we considered spatial semi-discretization by a standard nite element method. We proved a result analogous to that of Beyn including optimal order error bounds in both the L2 and H 1 norms. Noting that the analysis in [12] is rather ad hoc, and that the more general framework in [1] does not readily apply to spatial discretizations, we decided to reconsider this problem. In x2 below we provide an abstract framework for the long-time aspects of the analysis, which is based on carefully chosen assumptions to be checked in each application by proving rather standard nite-time error and perturbation estimates. More precisely, in x2.1 we prove a shadowing result for discrete dynamical systems of the form un+1 = S (un ), where S is a nonlinear operator in a Banach space X . We assume that S = L + N , where the bounded linear operator L is hyperbolic, and the nonlinear remainder N has a small Lipschitz constant on a subset D  X . This is an adaptation of the classical shadowing lemma of Anosov [2] and Bowen [4]. In x2.2 the mapping S is studied together with a family of approximations Sh = Lh + Nh , where Lh is linear, and it is assumed that we have access to bounds for Lh ? L and Sh ? S , as well as estimates of the Lipschitz constant of Nh. The main result of x2.2 is a theorem analogous to that of Beyn concerning the behavior of the orbits of S and Sh near a hyperbolic xed point of S . If S (t;
) is a continuous dynamical system (nonlinear semigroup) with orbits u(t) = S (t; u0 ), t  0, u(0) = u0 , then, for xed T , the mapping S = S (T;
) de nes a discrete dynamical system with orbits un = u(nT ), n = 0; 1; 2; : : : , satisfying the assumptions of x2.1 in a neighborhood D of a hyperbolic xed point. In x3 we apply the abstract framework in the context of a system of reactiondiusion equations discretized in the spatial variables by a standard nite element method, and in the time variable by means of the backward Euler method. The assumptions on Lh ? L and Sh ? S are veri ed by application of rather standard error estimates over the nite time interval [0; T ], which we quote from Larsson [11]. Our framework is similar to that of [1] but more exible. First of all it admits applications with both time and space discretization. In the applications discussed in x3 it also allows us to obtain error bounds of optimal order in both the L2 and H 1 norms. Moreover, it avoids the assumption that Sh ? S is small in C 1 (D; X ) that was used in [1], but which we found inconvenient. Note, in this connection, that we do not assume that L is a derivative of S . This is important, because even in a situation where L is formally a linearization of S , it may not be a Frechet derivative with respect to the norms that we use, see Remark 3 below. If X; Y are Banach spaces, then L(X; Y ) denotes the space of bounded linear operators from X into Y , L(X ) = L(X; X ), and BX (x; ) denotes the closed ball in X with center x and radius .

A SHADOWING RESULT

3

2. A general framework 2.1. A basic shadowing result. We consider a mapping S : D  X ! X , where X is a Banach space and D a nonempty subset of X . It is assumed that S can be decomposed in the form (2.1) S =L+N in a such a way that, for some constants   1 and  2 (0; 1), the following hypotheses (HL) and (HN) are ful lled. Hypothesis (HL). L 2 L(X ), i.e., L is a bounded linear operator in X . Furthermore, X can be decomposed as a direct sum X = X1  X2 of closed subspaces X1 , X2 that are invariant by L, i.e., LXi  Xi , i = 1; 2. If Li 2 L(Xi ), i = 1; 2, denotes the restriction of L to Xi , then L1 is invertible and kL?1 1 kL(X1 )  ; kL2 kL(X2 )  : (2.2) Moreover, the projections Pi , i = 1; 2, associated with the decomposition X = X1  X2 (i.e., Pi x = xi , i = 1; 2, for x = x1 + x2 , xi 2 Xi ) satisfy (2.3) kPi kL(X )  : Hypothesis (HN). The mapping N : D ! X is Lipschitz continuous with a Lipschitz constant that satis es Lip(N )  1 4? : (2.4) Note that the boundedness of the projections P1 ; P2 is a consequence of the closedness of subspaces X1 ; X2 ; this is a well-known consequence of the closed graph theorem, see [10, p. 167]. We now state the main result of this section. Theorem 2.1. (i) Assume that for the mapping S in (2.1) the hypotheses (HL), (HN) are satis ed and set (2.5)  := 1 4? : Let i and f be integers, i < f , and let fx~n gfn=i  D be a sequence. If fxn gfn=i  D is an orbit of S , i.e., xn+1 = S (xn ), n = i; : : : ; f ? 1, which satis es the boundary conditions (2.6) P2 xi = P2 x~i ; P1 xf = P1 x~f ; then (2.7) sup kxn ? x~n k   sup kx~n+1 ? S (~xn )k: inf

inf ?1

(ii) Assume, in addition to (HL), (HN), that the domain D of S contains a closed ball BX (z; ) and that (2.8) kz ? S (z )k  =:

Then, for any sequence fx~n gfn=i  BX (z; =()), there exists an orbit fxn gfn=i  BX (z; ) of S for which (2.6) and hence (2.7) hold.

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STIG LARSSON AND J.-M. SANZ-SERNA

Remark 1. With the terminology used in shadowing theory (see, e.g., [6]) we say that fx~n gfn=i is a -pseudo-orbit of S if supinf ?1 kx~n+1 ? S (~xn )k  . If this is the case, then the estimate (2.7) means that fx~n gfn=i is -shadowed by the true orbit fxn gfn=i with shadowing distance   . Part (ii) ensures the existence of a shadow orbit. Remark 2. Analogous results hold also for in nite sequences fx~n gfn=?1 , fx~n g1 n=i , f or fx~n g1 with obvious modi cations. For instance, for a sequence f x ~ g n n=?1 , n=?1 the rst condition in (2.6) is absent and the ranges in (2.7) become n  f and n  f ? 1. The proof is essentially the same as that given below for nite sequences. Note in this connection that the stability constant  in (2.7) does not depend on the initial and nal indices i and f . The theorem is proved by using the following lemmas. Lemma 2.2. Assume that L satis es hypothesis (HL), and that i; f are integers with i < f . Set  = f ? i, X = X  +1 , Y = X2  X   X1 , and de ne a linear operator L : X ! Y by L : (xi ; : : : ; xf ) 7! (yi ; : : : ; yf +1 ), where (2.9) yi = P2 xi ; yf +1 = P1 xf ; yn+1 = xn+1 ? Lxn ; n = i; : : : ; f ? 1: Then L is invertible and, with respect to the supremum norm of the product spaces X; Y, kL?1 kL(Y;X)  1 2? : (2.10) Proof. Given an element y = (yi ; : : : ; yf +1 ) 2 Y, we de ne x = (xi ; : : : ; xf ) 2 X, by the relations xn = P1 xn + P2 xn , where

P1 xn = (L?1 1 )f ?n yf +1 ? P2 xn = (L2 )n?i yi +

n X

f X

j =n+1

j =i+1

(L?1 1 )j?n P1 yj ;

(L2 )n?j P2 yj :

It is a simple matter to check that (2.9) holds. This proves that L is onto. To see that L is one-to-one, assume that yn , n = i; : : : ; f + 1, in (2.9) vanish. Then, by (2.9), P2 xi = 0 and P1 xf = 0. Recursion in (2.9) reveals that P2 xn = 0 for n = i + 1; : : : ; f . Similarly, a descending recursion in (2.9) shows that P1 xn = 0 for n = f ? 1; : : : ; i, so that the kernel of L is trivial. To derive (2.10), use (2.2){(2.3) in the de nition of xn , n = i; : : : ; f ,

kxn k  f ?nkyk + 

 1+ 

1 X j =n

f X j =n+1

f X

j =n+1

j?n kyk + n?i kyk +

j?n + n?i +

j?n +

n X j =?1

n X j =i+1

n X

j =i+1



n?j kyk

n?j kyk = 1 2? kyk: 

n?j kyk

A SHADOWING RESULT

5

Lemma 2.3. Let X and Y be Banach spaces and let L : X ! Y be a linear bijection with bounded inverse L?1 . Assume that the mapping N : D  X ! Y, de ned in a nonempty subset D of X, is Lipschitz continuous with

 := kL?1 kL(Y;X) Lip(N) < 1: De ne S = L + N and set  := kL?1 kL(Y;X)=(1 ? ). Then (2.11) kx1 ? x2 kX  kS(x1 ) ? S(x2 )kY ; for all x1 ; x2 2 D: Proof. The bound (2.11) follows readily from the identity

x1 ? x2 = L?1 ?S(x1 ) ? S(x2 )
? L?1 ?N(x1) ? N(x2 )
:

Lemma 2.4. In addition to the hypotheses of Lemma 2.3, assume that the domain D of N contains a closed ball BX (z; ). Then, for each y 2 Y with (2.12) ky ? S(z)kY  =; the equation S(x) = y has a unique solution x 2 BX (z; ). Proof. This is a simple consequence of the contraction mapping theorem. In fact, if we de ne T(x) = L?1 (y ? N(x)), then Lip(T) =  < 1. Moreover, the identity ?




?

T(x) ? z = L?1 y ? S(z) ? L?1 N(x) ? N(z) and (2.12) imply that T maps BX (z; ) into itself.




Proof of Theorem 2.1. Given i and f , we construct the spaces X and Y and the linear operator L as in Lemma 2.2. We further consider the mapping N : D = D +1  X ! Y de ned by N(x) = y, where

yi = yf +1 = 0; yn+1 = ?N (xn ); n = i; : : : ; f ? 1: The assumption (HN) implies that Lip(N)  (1 ? )=(4) and (2.10) leads to kL?1 k Lip(N)  1=2. We can therefore apply Lemma 2.3 with  = 1=2; this yields a value of  = kL?1 k=(1 ? ) that coincides with  in (2.5). Note also that S is

(2.13)

de ned by S(x) = y, where (see (2.1), (2.9), and (2.13))

yi = P2 xi ; yf +1 = P1 xf ; yn+1 = xn+1 ? S (xn ); n = i; : : : ; f ? 1: The estimate (2.7) is then a straightforward consequence of (2.11). To prove part (ii) of the theorem, we apply Lemma 2.4 in the ball B (z; ) = B (z; ) +1 , where z = (z; : : : ; z ). Given fx~n gfi=1  B (z; =()), we put y = (P2 x~i ; 0; : : : ; 0; P1x~f ). The condition (2.12) is satis ed. In fact, the rst component of y ? S(z) is the vector P2 x~i ? P2 z , whose norm can be estimated by kP2 k kx~i ? z k  =. Similarly, the last component of y ? S(z) has norm  =. The remaining components of y ? S(z) equal 0 ? (z ? S (z )) and, in view of (2.8), are also bounded in norm by =. Since (2.12) is satis ed, the equation S(x) = y has a solution x = (xi ; : : : ; xf ). The choice of y ensures that fxngfn=i is the sequence required.

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2.2. Shadowing and approximation. We now consider, along with the mapping S in (2.1), a family of approximations fSh g. Let H be a set of positive numbers with inf H = 0. For each h 2 H, let Xh be a subspace of the Banach space X , in such a way that there exist bounded projections Qh : X ! Xh and a number  1 with (2.14) kQhkL(X )  : We assume that the spaces Xh approximate X in the sense that (2.15) lim Q u = u; for all u 2 X: h!0 h For h 2 H, we consider mappings Sh : Dh  Xh ! Xh with domains Dh = D \ Xh , that approximate S in the sense that a continuous positive function (h) exists such that (2.16) lim (h) = 0 h!0 and (2.17) kSh (Qh u) ? S (u)k  (h); for all u 2 D such that Qh u 2 Dh . Finally, we assume that Sh can be decomposed as (2.18) Sh = Lh + Nh ; and that (HL) and (HN) hold for this decomposition. More precisely, this means that Lh 2 L(Xh ); Xh can be decomposed as a direct sum Xh = X1h  X2h of closed subspaces invariant by Lh; the restrictions Lih , i = 1; 2, of Lh to Xih satisfy (2.19) kL?1h1kL(X1h )  ; kL2hkL(X2h )  ; the associated projections satisfy (2.20) kPih kL(X )  ; i = 1; 2; and Nh : Dh ! Xh with Lipschitz constant Lip(Nh )  1 4? : Note that, Xih is in general dierent from Xi \ Xh; the latter may well be the trivial subspace f0g. Theorem 2.5. (i) Assume that the subspaces Xh of the Banach space X possess the approximation properties (2.14){(2.15), the mappings S in (2.1) and Sh in (2.18) satisfy (HL) and (HN), and the Sh approximate S as in (2.17). Let i and f be integers with i < f . Then the following results hold. (i.a) Let fuh;ngfn=i  Dh be an orbit of Sh . If fungfn=i is an orbit of S with (2.21) P2 ui = P2 uh;i; P1 uf = P1 uh;f ; then (2.22) sup kun ? uh;nk  (h); inf

where  is the constant in (2.5). (i.b) Let fungfn=i  D be an orbit of S with (2.23) Qh un 2 Dh ; n = i; : : : ; f:

A SHADOWING RESULT

7

If fuh;ngfn=i  Dh is an orbit of Sh with (2.24) P2h uh;i = P2h Qh ui ; P1h uh;f = P1h Qhuf ; then (2.25) sup kQhun ? uh;n k  (h): inf

(ii) Assume that the hypotheses in (i) hold and that S has a xed point u such that BX (u; )  D for some  > 0. Then the following conclusions hold. (ii.a) For any orbit fuh;ngfn=i  BX (u; 0 ), 0 = =() < , of Sh there is an orbit fungfn=i of S for which (2.21) and (2.22) are true. (ii.b) Let h be small enough for the inequalities (2.26) ku ? Qhuk  =2; 2 (h)   to hold (cf. (2.15){(2.16)). Then, for any orbit fun gfn=i  BX (u; 0 ), 0 = =(2 ), of S , the relation (2.23) is satis ed and there is an orbit fuh;ngfn=i of Sh for which (2.24) and (2.25) hold. (ii.c) For h chosen as in (ii.b), the mapping Sh has a xed point that is unique in the ball BXh (Qh u; =2). Furthermore, (2.27) kQh u ? uhk  (h): Proof. To prove (i.a) we apply part (i) of Theorem 2.1 with x~n = uh;n 2 Dh  D, and xn = un . Then (2.7) yields sup kun ? uh;n k   sup kuh;n+1 ? S (uh;n )k: inf inf ?1 Since uh;n+1 ? S (uh;n) = Sh (Qh uh;n) ? S (uh;n), the bound (2.22) is a consequence

of (2.17). For part (i.b) we again resort to part (i) of Theorem 2.1, but this time with Sh playing the role of S , and x~n = Qh un 2 Dh , xn = uh;n 2 Dh . The estimate (2.7) reads sup kuh;n ? Qh un k   sup kQhun+1 ? Sh(Qh un )k; inf

inf ?1

and (2.25) is a consequence of (2.14), (2.17), in view of the identity Qh un+1 ? Sh (Qh un ) = Qh[S (un ) ? Sh (Qh un)]: Part (ii.a) is a direct consequence of part (ii) of Theorem 2.1 with z = u. For (ii.b) we use part (ii) of Theorem (2.1) with Sh playing the role of S and z = Qhu. If v 2 BXh (Qh u; =2), then the assumption (2.26) implies kv ? uk  kv ? Qh uk + kQhu ? uk  ; this shows that BXh (Qh u; =2) is contained in BX (u; ), which in turn is assumed to be contained in D. Hence BXh (Qh u; =2)  Dh. Furthermore, by (2.14), (2.17) and (2.26), (2.28) kQ u ? S (Q u)k = kQ [S (u) ? S (Q u)]k  (h)  =2 ; h

h

h

h

h

h



so that (2.8) holds with the role of BX (z; ) played by BXh (Qh u; =2). If fun gfn=i  BX (u; 0 ), then, by (2.14), fQhun gfn=i  BXh (Qh u; =(2))  BXh (Qh u; =2) and (2.23) holds. The existence of an orbit fuh;ngfn=i of Sh satisfying (2.24) now follows from Theorem 2.1.

8

STIG LARSSON AND J.-M. SANZ-SERNA

For (ii.c) we apply Lemma 2.4 with X = Y = Xh, L = Lh ? I , N = Nh, y = 0 and the role of BX(z; ) played by BXh (Qhu; =2), which we know to be contained in Dh . By Lemma 2.8, k(Lh ? I )?1 k  2=(1 ? ) leading in Lemma 2.3 to  = 1=2 and a value of  that agrees with the value in (2.5). The condition (2.12) is ful lled in view of (2.28), because y ? S(z) = ?Sh(Qh u) + Qh u. It only remains to prove the bound (2.27). We apply again Lemma 2.4, but this time in the smaller ball BXh (Qh u;  (h))  BXh (Qhu; =2); the condition (2.12) is still

ful lled in view (2.28), so that the unique xed point uh of Sh in BXh (Q)h u; =2) also lies in BXh (Qh u;  (h)). Our next theorem gives a condition under which the \hyperbolicity" (HL) of the operator L carries over to Lh. Theorem 2.6. Assume that the operator L 2 L(X ), X a Banach space, satis es hypothesis (HL) and choose ~ 2 (; 1), ~ > . Assume that the subspaces Xh with corresponding projections Qh are such that (2.14) holds and that the operators Lh 2 L(Xh ) approximate L in the sense that (2.29) kL ? Lh QhkL(X )  (h) with (h) as in (2.16). Then there exists h0 > 0 (depending only on , , , ~, ~, and the function ) such that, uniformly for h < h0 , the operators Lh satisfy (HL) with constants ~, ~. For the proof of the theorem we need two simple lemmas. Lemma 2.7. Let A; B 2 L(X ), X a Banach space, with A?1 2 L(X ) and kBk  1 kA?1 k?1 . Then (A + B )?1 2 L(X ) and 2 k(A + B )?1 ? A?1 kL(X )  2kA?1k2L(X )kB kL(X ) : P ?1 n ?1 Proof. We have k(A + B )?1 kL(X ) = kA?1 1 n=0 (?A B ) kL(X )  2kA kL(X ) ? 1 ? 1 ? 1 ? 1 and (A + B ) ? A = ?(A + B ) BA . Lemma 2.8. Assume that the operator L 2 L(X ), X a Banach space, satis es hypothesis (HL) and let ! be a complex number with j!j = 1. Then (!I ? L)?1 2 L(X ) with k(!I ? L)?1 kL(X )  1 2? : Proof. Let j!j = 1. Assumption (2.2) implies that (!I ? L1 )?1 = ?L1 and hence

1 X

n=0

(!L?1 1 )n ; (!I ? L2)?1 = !?1

1 X

n=0

(!?1 L2 )n ;

k(!I ? L1 )?1 kL(X1 )  1 ?  ; k(!I ? L2 )?1 kL(X2 )  1 ?1  : Using also (2.3) and (!I ? L)?1 = (!I ? L1)?1 P1 + (!I ? L2 )?1 P2 , we obtain k(!I ? L)?1 kL(X )  11 +?    1 2? :

A SHADOWING RESULT

9

Proof of Theorem 2.6. De ne L~ h = Lh Qh 2 L(X ) and let j!j = 1. We begin by showing that (!I ? L~ h )?1 2 L(X ). In order to do so we shall apply Lemma 2.7 with A = !I ? L and B = L ? L~ h. From Lemma 2.8 we know that k(!I ? L)?1k  2=(1 ? ). Hence, for h suciently small, we have kL ? L~ hkL(X )  (h)  1 4?  2k(!I ?1 L)?1 k : Now Lemma 2.7 applies and gives (!I ? L~ h )?1 2 L(X ) and 2  (2.30) k(!I ? L~ h)?1 ? (!I ? L)?1 kL(X )  2 1 2? kL ? L~ h kL(X )  K(h); where K = 2(2=(1 ? ))2 . We next show that (!I ? Lh )?1 2 L(Xh ) and that (2.31) (!I ? Lh)?1 Qh = (!I ? L~ h )?1 Qh : In fact, if f 2 X , then u = (!I ? L~ h )?1 Qh f 2 X satis es Qh f = (!I ? L~ h )(Qh u + (I ? Qh)u) = (!I ? Lh)Qh u + !(I ? Qh)u: We conclude that (I ? Qh )u = 0, and (!I ? Lh )Qh u = Qhf , so that u = Qh u = (!I ? Lh )?1 Qh f , which implies (2.31). We have now proved that, for each suciently small h, Lh has no spectrum on the unit circle. By a standard theorem, see, e.g., [10, Theorem III{6.17, p. 178], this implies the existence of a splitting Xh = X1h  X2h as required in assumption (HL). It only remains to prove that the corresponding inequalities (2.19) and (2.20) with constants ~, ~ hold uniformly with respect to h. In order to obtain a bound for kP2h kL(Xh ) we rst estimate k(P2 ? P2h )Qh kL(X ). Using the representations Z Z P2 = 21i (!I ? L)?1 d!; P2h = 21i (!I ? Lh)?1 d!; ? ? where ? denotes the unit circle with positive orientation, together with (2.31), (2.30), and (2.14) we obtain

Z ?
k(P2 ? P2h )Qh kL(X ) = 21

(!I ? L)?1 ? (!I ? L~ h)?1 d! Qh

L(X )  K(h): ? This implies that, for x 2 Xh , ?
kP2h xk  kP2 xk + k(P2 ? P2h )xk   + K(h) kxk; so that kP2h kL(Xh )   + K(h)  ~; provided that h is suciently small. Since (P1 ? P1h )Qh = (P2h ? P2 )Qh , we also have kP1h kL(Xh )  ~. We now turn to the bound for kL2h kL(X2h ) . Since Z Z L2 P2 = 21i !(!I ? L)?1 d!; L2h P2h = 21i !(!I ? Lh )?1 d!; ? ? we have

Z


? k(L2 P2 ? L2h P2h )Qh kL(X ) = 21

! (!I ? L)?1 ? (!I ? L~ h )?1 d! Qh

L(X ) ?  K(h):

10

STIG LARSSON AND J.-M. SANZ-SERNA

Hence, for x 2 X2h , kL2hxk = kL2h P2h xk  k(L2hP2h ? L2 P2 )Qh xk + kL2P2 xk  K(h)kxk + kP2xk: Here, since Qh x = P2h x = x, kP2 xk  k(P2 ? P2h )Qh xk + kP2h xk  K(h)kxk + kxk; so that ?
kL2hxk   + 2 K(h) kxk; and we conclude that, for small h, kL2hkL(X2h )   + 2 K(h)  ~: The required bound for kL?1h1 kL(X1h ) is obtained in the same way, using the representations Z Z L?1 P1 = ? 21i !?1 (!I ? L)?1 d!; L?1h1 P1h = ? 21i !?1 (!I ? Lh )?1 d!: ? ? 3. Application to a system of reaction-diffusion equations The purpose of this section is to illustrate how the theory of the previous section can be applied in the context of a standard nite element approximation of a system of reaction-diusion equations. 3.1. The continuous problem. We consider the model problem ut ? D
u = f~(u); x 2 ; t > 0; u = 0; x 2 @ ; t > 0; (3.1) u(
; 0) = u0 ; x 2 ; where

is a bounded domain in Rd, d = 1; 2; 3, u = u(x; t) 2 Rs , ut = @u=@t,
u = P 2 2 i @ u=@xi , D = diag(d1 ; : : : ; ds ) is a diagonal matrix of constant coecients di > 0, and f~ : Rs ! Rs is continuously dierentiable. We assume that is either a convex polygon or has a smooth boundary. If d = 2; 3 we assume, in addition, that the Jacobian of f~ satis es the growth condition jf~0 ( )j  C (1 + j j );  2 Rs ; where j
j denotes the Euclidean norm on Rs and the induced matrix norm, and where  = 2 if d = 3,  2 [0; 1) if d = 2. In the sequel we use the Hilbert space H = (L2 ( ))s , with its standard norm k
k and inner product (
;
). The norms in the Sobolev spaces (H m ( ))s , m  0 are denoted by k
km . The space V = (H01 ( ))s , with norm k
k1, consists of the functions in (H 1 ( ))s that vanish on @ . We de ne the operator A = ?D
with domain D(A) = (H 2 ( ) \ H01 ( ))s . Then A is a closed, densely de ned and selfadjoint operator in H with compact inverse. Moreover, our assumptions guarantee that the mapping f~ induces an operator f : V ! H through f (v)(x) = f~(v(x)), see Lemma 3.1 below. The initial-boundary value problem (3.1) may then be formulated as an initial value problem in V : (3.2) u0 + Au = f (u); t > 0; u(0) = u0 :

A SHADOWING RESULT

11

We assume further that (3.2) has a stationary solution u with u 2 D(A), Au = f (u); by standard embedding results u is continuous in the closure of . The formula (Bv)(x) = f~0 (u(x))v(x) clearly de nes an operator B 2 L(H ). The operator A = A ? B is the linearization of A ? f at u and, being a bounded perturbation of A, it is a sectorial operator in H (see [9, Theorem 1.3.2]). Hence ?A is the generator of an analytic semigroup e?tA . We assume that u is \hyperbolic", i.e., that the spectrum of A does not intersect the imaginary axis. Let P1 and P2 the projections respectively associated with the sets 1 = (A) \ fRe z < 0g and 2 = (A) \ fRe z > 0g that partition the spectrum (A) of A and let H1 and H2 be the ranges of P1 and P2 . It follows that H is a direct sum H = H1  H2 ; the subspaces Hi are invariant under A and, if Ai , i = 1; 2, denotes the restriction of A to Hi , then A1 2 L(H1 ), D(A2 ) = D(A) \ H2 . Furthermore, there are M  1 and  > 0, such that, t  0; v 2 H1 ; m = 1; 2; ke?tA1 vkm  Met kvk; ? m= 2 ? t ? t A 2 (3.3) ke vkm  Mt e kvk; t > 0; v 2 H2 ; m = 1; 2; t  0; v 2 H2 \ V: ke?tA2 vk1  Me?tkvk1 ; We refer to [9, x1.5] for these facts. Since H1  D(A), we see that we also have a direct sum V = V1  V2 , where V1 = H1 and V2 = H2 \ V , with associated projections P1 jV and P2 jV . By the closed graph theorem, we may select a constant   1 such that (3.4) kPi kL(H )  ; kPi kL(V )  ; i = 1; 2: By combination of these with (3.3) we have ke?tA vk1  Ct?1=2 et kvk; t > 0; v 2 H; (3.5) ke?tA vk1  Cet kvk1; t  0; v 2 V: With F (v) = f (v) ? Bv, we may rewrite (3.2) as (3.6) u0 + Au = F (u); t > 0; u(0) = u0 ; and clearly we also have (3.7) Au = F (u): As shown by the following lemma, whose proof is similar to that of Lemma 2.2 in [12], the nonlinear operator F : V ! H is Lipschitz continuous with a Lipschitz constant that may be rendered arbitrarily small by restricting the attention to a suciently small neighborhood of u. Lemma 3.1. If v; w 2 BV (u; ), then (3.8) kF (v) ? F (w)k  k()kv ? wk1 ; where k() = O() as  ! 0. The initial value problem (3.6) (or (3.2)) has a unique local solution for any initial datum u0 2 V , see [9, Theorem 3.3.3]. We denote by S (t;
) the corresponding (local) solution operator, so that u(t) = S (t; u0 ) is the solution of (3.6). The following lemma shows that the local solutions can be extended in time, if they start suciently near u. Lemma 3.2. For each 1 > 0 and T > 0 there is  > 0 such that, if u0 2 BV (u; ), then S (t; u0 ) is de ned and belongs to BV (u; 1 ) for t 2 [0; T ].

12

STIG LARSSON AND J.-M. SANZ-SERNA

Proof. Let 1 ; T > 0 be given. For  > 0 let  2 [0; T ] be the largest time such that u0 2 BV (u; ) implies that u(t) = S (t; u0 ) exists and belongs to BV (u; 1 + 1) for t 2 [0;  ]. We must choose  such that  = T . Let z (t) = u(t) ? u. Forming the dierence between (3.6) and (3.7) and using the variation of constants formula, we obtain

z (t) = e?tA z (0) +

t

Z

0

?




e?(t?s)A F (u(s)) ? F (u) ds:

Invoking (3.5) and (3.8), we therefore have, for t 2 [0;  ],

kz (t)k1  Cet kz (0)k1 + C  CeT

Z

t

0



 + k(1 + 1)

(t ? s)?1=2 e(t?s) kF (u(s)) ? F (u)k ds Z

0

t



(t ? s)?1=2 kz (s)k1 ds :

Gronwall's lemma (see [13, Lemma 5.6.7] or [9, Exercise 4 of x6.1]) now yields kz (t)k1  C (1 ; T ); t 2 [0;  ]; so that if we choose  = 1 =C (1 ; T ), then kz (t)k1  1 ; t 2 [0;  ]: If  < T , then by local existence we obtain a contradiction with the maximality of  . Hence, S (t; u0 ) is de ned and belongs to BV (u; 1 ) for t 2 [0; T ]. The following lemma provides a bound for the H 2 norm of the solution found in Lemma 3.2. Lemma 3.3. Let 1 > 0, T > 0 and assume that u(t) = S (t; u0) exists and belongs to BV (u; 1 ) for t 2 [0; T ]. Then there exists C (1 ; T ) such that ku(t)k2  C (1 ; T ) t?1=2 ; t 2 (0; T ]: Proof. In view of (3.8) we have kF (u(t))k  C (1 ) for t 2 [0; T ]. The proof is now obtained by tracing the constants in [9, Theorem 3.5.2]. In order to set the present problem in the framework of x2, we choose T such that (3.9)  := Me?T < 1; where M and  are the constants in (3.3). We then de ne S = S (T;
), L = e?T A , N = S ? L. It is clear from the above that assumption (HL) is satis ed, both with X = H and X = V . In order to choose the domain D so that (HN) holds we need the following result. Lemma 3.4. For each  > 0 there is  > 0 such that v1; v2 2 BV (u; ) implies kN (v1 ) ? N (v2 )km  kv1 ? v2 km ; m = 0; 1: Proof. Let T be as in (3.9) and let 1 > 0. We rst carry out an a priori estimation under the assumption that ui = S (t; vi ), i = 1; 2, exist and belong to BV (u; 1 ) for t 2 [0; T ]. From the variation of constants formula

ui (t) = e?tA vi +

Z

0

t

e?(t?s)A F (ui (s)) ds; t 2 [0; T ];

A SHADOWING RESULT

so that N (vi ) = wi (T ), where

wi (t) =

t

Z

Using (3.5) and (3.8), we obtain

kw1 (t) ? w2 (t)k1  Cet

0

e?(t?s)A F (ui (s)) ds: t

Z

0

13

(t ? s)?1=2 kF (u1(s)) ? F (u2 (s))k ds t

Z

 C (T )k(1 ) (t ? s)?1=2 ku1(s) ? u2 (s)k1 ds: 0

Here u1 (s) ? u2 (s) = e?sA (v1 ? v2 ) + w1 (s) ? w2 (s), so that another application of (3.5) yields, for t 2 [0; T ], kw1 (t) ? w2 (t)k1  C (T )k(1 )kv1 ? v2 k + C (T; 1 )

t

Z

0

(t ? s)?1=2 kw1 (s) ? w2 (s)k1 ds:

Gronwall's lemma now shows that (3.10) kw1 (t) ? w2 (t)k1  C (T; 1 )k(1 )kv1 ? v2 k; t 2 [0; T ]: This is the required a priori bound and we may now complete the proof. Let  > 0. Since k(1 ) = O(1 ) and C (T; 1 ) = O(1) as 1 ! 0, we may choose 1 so that C (T; 1 )k(1 )  . Lemma 3.2 provides  such that v1 ; v2 2 BV (u; ) implies u1(t); u2 (t) 2 BV (u; 1 ) for t 2 [0; T ], and (3.10) then yields kw1 (T ) ? w2 (T )k1  kv1 ? v2 k; which implies both the required estimates. Lemma 3.4 shows that there is  such that, if we set D = BV (u; ), then N satis es (HN) with both X = H and X = V . Moreover, we have found a larger radius 1 >  such that (3.11) u0 2 D = BV (u; ) ) S (t; u0) 2 BV (u; 1 ); t 2 [0; T ]: In summary, we have chosen the parameters so as to make sure that S = L + N satis es (HL) and (HN) in both X = H and X = V . Remark 3. Note that L is the linearization of S at u. In fact, the mapping S : D  V ! V is Frechet dierentiable with L = S 0 (u) 2 L(V ). However, the mapping S : D  H ! H is not dierentiable, because D = BV (u; ) is not a neighborhood of u with respect to the topology of H . 3.2. The discrete problem. In this section we rst discretize the initial-boundary value problem (3.1) with respect to the spatial variables by means of a standard piecewise linear nite element method and apply the shadowing results of x2.2. At the end of the section we then brie y discuss completely discrete approximations obtained by means of the backward Euler time-stepping. Let fVh g0

0; uh (0) = u0h;

which is the discrete analogue of (3.2). In the same way as for the continuous problem we can show that there is a local solution operator Sh (t;
), such that uh(t) = Sh(t; u0h ) is the unique local solution of (3.14). Just as in the continuous case the proof is based on the variation of constants formula, the analyticity of the semigroup exp(?tAh ), and the local Lipschitz condition for the mapping f : V ! H , see [11]. With Ah = Ah ? Qh B and F (v) = f (v) ? Bv as before, we rewrite (3.14) as u0h + Ah uh = Qh F (uh ); t > 0; uh (0) = u0h; which is the discrete version (3.6). Since Ah is selfadjoint, positive de nite (uniformly in h), and QhB is bounded, we deduce that Ah is sectorial (uniformly in h), so that for some c > 0 ke?tAh vk1  Cect kvk1 ; t  0; v 2 Vh ; (3.15) ? t A ? 1 = 2 ct ke h vk1  Ct e kvk; t > 0; v 2 Vh ; which are discrete versions of (3.5). Here we have employed the equivalence of norms kvk1  kA1h=2 vk for v 2 Vh . The inequalities (3.15) can also be proved by noting that uh (t) = e?tAh u0h satis es (3.14) with f (uh) replaced by Buh , and by making estimations based on the variation of constants formula. From [11] we quote the following a priori error estimates. Lemma 3.5. Let 0 <   T and assume that S (t; u0); Sh(t; u0h) 2 BV (u; ) for t 2 [0;  ]. Then, for t 2 [0;  ], we have   kSh (t; u0h ) ? S (t; u0 )k  C (; T ) ku0h ? Qh u0 k + h2 t?1=2 ; 



kSh (t; u0h) ? S (t; u0 )k1  C (; T )t?1=2 ku0h ? u0k + h :

A SHADOWING RESULT

15

The following is a discrete analogue of Lemma 3.2. Lemma 3.6. For each 1 > 0, T > 0 there are  > 0, h0 > 0 such that, if u0h 2 BV (u; ) \ Vh and h < h0 , then Sh(t; u0h ) exists and belongs to BV (u; 1 ) \ Vh for t 2 [0; T ]. Proof. Let 1 ; T > 0 be given. For  > 0 let  2 [0; T ] be the largest time such that u0h 2 BV (u; )\Vh implies that Sh (t; u0h ) exists and belongs to BV (u; 1 +1)\Vh for t 2 [0;  ]. By local existence there are t0 > 0 and  > 0 such that u0h 2 BV (u; )\Vh implies that Sh (t; u0h) exists and belongs to BV (u; 1 ) \ Vh for t 2 [0; t0 ]. Moreover, the second error estimate of Lemma 3.5 gives the a priori estimate 



kSh (t; u0h ) ? uk1  C (1 ; T )t?1=2 ku0h ? uk + h  C (1 ; T )t?0 1=2( + h0 )  1 ; t 2 [t0 ;  ];

provided that  and h0 are suciently small. If  < T , then by local existence we obtain a contradiction with the maximality of  . Hence, Sh (t; u0 ) is de ned and belongs to BV (u; 1 ) \ Vh for t 2 [0; T ]. We will also use the error bounds (3.16)

ke?tAh Qh v ? e?tA vk  Ch2 t?1 et kvk; t > 0; v 2 H; ke?tAh Qh v ? e?tA vk1  Cht?1=2 et kvk1; t > 0; v 2 V;

which can be proved by the using the techniques of [11]. With T as in (3.9) we de ne Lh = e?T Ah , and (3.16) shows kLhQh ? LkL(H ) + hkLhQh ? LkL(V )  C (T )h2 : The assumption (2.29) is thus satis ed with both X = H and X = V . We conclude that Theorem 2.6 applies, showing that, for small h, Lh satis es (HL) with slightly larger constants ~ > , ~ > . Adjusting , , we may conclude that (2.19), (2.20) hold. Finally, we de ne Sh = Sh (T;
), Nh = Sh ? Lh , and note that, after these preparations, the analogue of Lemma 3.4 holds with the same proof. As for the continuous problem we may select , h0 such that, for h < h0 , Nh satis es (HN) with Dh = D \ Vh , D = BV (u; ). The argument also selects 1 such that, in analogy with (3.11), (3.17) u0h 2 Dh ) Sh (t; u0h ) 2 BV (u; 1 ); t 2 [0; T ]: Moreover, using Lemma 3.5 together with (3.11) and (3.17), we see that, if v 2 D, Qhv 2 Dh , then kSh (Qh v) ? S (v)k  C (1 ; T )h2 ; kSh(Qh v) ? S (v)k1  C (1 ; T )h; since kQhv ? vk  Chkvk1  Ch. We conclude that (2.17) holds with X = H and (h) = Ch2 , and with X = V and (h) = Ch. We have now checked all the assumptions of Theorem 2.5 and we are ready to apply it. Theorem 3.7. There are positive numbers 0, h0 , and C such that, for any h < h0, the following hold:

16

STIG LARSSON AND J.-M. SANZ-SERNA

(1) If uh is a solution of (3.12) with uh (t) 2 BV (u; 0 ) for t 2 [0; T ], then there is a solution u of (3.1) such that ?
(3.18) kuh(t) ? u(t)km  C 1 + t?1=2 h2?m ; t 2 (0; T ]; m = 0; 1: (2) Conversely, if u is a solution of (3.1) with u(t) 2 BV (u; 0 ) for t 2 [0; T ], then there is a solution uh of (3.12) such that (3.18) holds. (3) Equation (3.12) has a stationary solution uh such that kuh ? uk1  Ch: Proof. Let , 1 , h0 , T be as above. Choose 0 and adjust h0 in such a way that the requirements of parts (ii.a) and (ii.b) of Theorem 2.5 are satis ed with X = V . (1) Let uh (t) 2 BV (u; 0 ) for t 2 [0; T ] and apply part (ii.a) with X = V to the sequence uh (nT ), nT 2 [0; T ], which is an orbit of Sh . This gives the existence of an orbit u(nT ), nT 2 [0; T ], of S , satisfying (2.21) of part (i.a), and hence (2.22) gives the special case m = 1 of the inequality (3.19) kuh(nT ) ? u(nT )km  Ch2?m ; nT 2 [0; T ]; m = 0; 1: Another application of part (i.a), now with X = H , proves the case m = 0 of (3.19). From the sequence u(nT ) we de ne u(t) = S (t ? nT; u(nT )) for t 2 [nT; (n +1)T ]. By uniqueness of solutions this is a solution of (3.1). Error bounds at intermediate times are obtained by combining (3.19) with Lemma 3.5 as follows. For t 2 [0; T ] we have   kuh(t) ? u(t)k1  C (1 ; T )t?1=2 kuh(0) ? u(0)k + h  C (1 ; T )t?1=2h: For t 2 [(n + 1)T; (n + 2)T ], n  0, we have   kuh(t) ? u(t)k1  C (1 ; 2T )t?1=2 kuh (nT ) ? u(nT )k + h  C (1 ; T )h: This proves the special case m = 1 of (3.18). The case m = 0 is obtained similarly. (2) Let u(t) 2 BV (u; 0 ) for t 2 [0; T ] and apply part (ii.b) with X = V to the sequence u(nT ), nT 2 [0; T ], which is an orbit of S . This gives the existence of an orbit uh(nT ), nT 2 [0; T ], of Sh , satisfying (2.24) of part (i.b), and hence (2.25) gives the special case m = 1 of the inequality (3.20) kQhu(nT ) ? uh (nT )km  Ch2?m ; nT 2 [0; T ]; m = 0; 1: Another application of part (i.b), now with X = H , proves the case m = 0 of (3.20). The required error bound (3.18) now follows as in part (1) above, noting that, by (3.13) and Lemma 3.3, kQhu(t) ? u(t)km  Ch2?m ku(t)k2  C (1 ; T )h2?mt?1=2 : (3) Part (ii.c) of Theorem 2.5 gives uh and the error bound is an immediate consequence of (2.27) and (3.13). We conclude this section by brie y indicating how time discretization by the backward Euler method can be incorporated into the above argument. After discretization with constant time steps k (3.14) becomes (Uj ? Uj?1 )=k + Ah Uj = Qhf (Uj ); tj = jk > 0; U0 = u0h : The local solution operator Sh;k (tj ; u0h ) is readily obtained by using the smoothing property of the corresponding linear evolution operator Eh;k (tj ) = (I ? kAh )?j , and the local Lipschitz condition for f : V ! H , see [11]. This smoothing property

A SHADOWING RESULT

17

carries over to the linearized operator Eh;k (tj ) = (I ? kAh)?j in the same way as in the semidiscrete case, see (3.15). Error bounds analogous to those of Lemma 3.5 can also be found in [11]. With these ingredients we may prove an analog of Lemma 3.6. Error bounds for Eh;k (tj ) analogous to those in (3.16) may be found in [11], and with a discrete time T suitably chosen we nd that Lh;k = Eh;k (T ) satis es (HL). Setting Sh;k = Sh;k (T;
), Nh;k = Sh;k ? Lh;k we then prove an analog of Lemma 3.4. Further arguments, parallel to those above, lead to an analog of Theorem 3.7 with an error bound of the form ?

? kUj ? u(tj )km  C 1 + t?j 1=2 h2?m + 1 + tj?(m+1)=2 k ; tj 2 (0; T ]; m = 0; 1: Remark 4. The framework of x2 applies also in the context of a nite element method for the Cahn-Hilliard equation, for which the nite time analysis was carried out in [8]. We skip the details. References [1] F. Alouges and A. Debussche, On the qualitative behavior of the orbits of a parabolic partial dierential equation and its discretization in the neighborhood of a hyperbolic xed point, Numer. Funct. Anal. Optim. 12 (1991), 253{269. [2] D. V. Anosov, Geodesic ows and closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). [3] W.-J. Beyn, On the numerical approximation of phase portraits near stationary points, SIAM J. Numer. Anal. 24 (1987), 1095{1113. [4] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Dieomorphisms, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 1975. [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [6] B. A. Coomes, H. Kocak, and K. J. Palmer, Rigorous computational shadowing of orbits of ordinary dierential equations, Numer. Math. 69 (1995), 401{421. [7] M. Crouzeix and V. Thomee, The stability in Lp and Wp1 of the L2 -projection onto nite element function spaces, Math. Comp. 48 (1987), 521{532. [8] C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a nite element method for the Cahn-Hilliard equation, Math. Comp. 58 (1992), 603{630. [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, 1981. [10] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1976. [11] S. Larsson, Nonsmooth data error estimates with applications to the study of the long-time behavior of nite element solutions of semilinear parabolic problems, IMA J. Numer. Anal., to appear. [12] S. Larsson and J.-M. Sanz-Serna, The behavior of nite element solutions of semilinear parabolic problems near stationary points, SIAM J. Numer. Anal. 31 (1994), 1000{1018. [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Dierential Equations, Springer-Verlag, New York, 1983. Department of Mathematics, Chalmers University of Technology and Goteborg University, S{412 96 Goteborg, Sweden

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Departamento de Matematica Aplicada y Computacion, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

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