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 finite element solutions of a reaction-diffusion 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 sufficiently small neighborhood of the equilibrium. Error bounds of optimal order in the L_{2} 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 fixed 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 fixed point.

The inspiration for this work came from an article of Beyn, [3], on multi-step approximations of systems of nonlinear ordinary differential equations, u′ = f(u). Beyn showed that if the continuous problem has a hyperbolic fixed point ū, then there is a neighborhood O of ū such that the following conclusion holds: for each initial value u_{0} ∈ O there is U_{0} ∈ O such that the approximate orbit U starting from U_{0} is close to the exact orbit u starting from u_{0} as long as the latter orbit stays in O. The error u − U satisfies an estimate, which is uniform with respect to u_{0} ∈ 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 U_{0} ∈ O there is u_{0} ∈ O such that the corresponding orbits U and u are optimally close as long as they remain in O. We emphasize that U_{0} ≠ u_{0} in general, because the initial value problem is typically unstable near a hyperbolic fixed point.

Date: January 1996.

1991 Mathematics Subject Classification. 65M15, 65M60.

Key words and phrases. Shadowing, semilinear parabolic problem, hyperbolic stationary point, finite element method, backward Euler, error estimate.

Published in Math. Comp. 68 (1999), 55–72.

The first author was partly supported by the Swedish Research Council for Engineering Sciences (TFR). The second author was partly supported by “Dirección General de Investigación Científica y Técnica” under project PB92–254.

Beyn’s result was extended to infinite 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 finite element method. We proved a result analogous to that of Beyn including optimal order error bounds in both the L_{2} 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 §2 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 finite-time error and perturbation estimates. More precisely, in §2.1 we prove a shadowing result for discrete dynamical systems of the form u_{n+1} = S(u_{n}), 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 §2.2 the mapping S is studied together with a family of approximations S_{h} = L_{h} + N_{h}, where L_{h} is linear, and it is assumed that we have access to bounds for L_{h} − L and S_{h} − S, as well as estimates of the Lipschitz constant of N_{h}. The main result of §2.2 is a theorem analogous to that of Beyn concerning the behavior of the orbits of S and S_{h} near a hyperbolic fixed point of S.

If S(t, ·) is a continuous dynamical system (nonlinear semigroup) with orbits u(t) = S(t, u_{0}), t ≥ 0, u(0) = u_{0}, then, for fixed T, the mapping S = S(T, ·) defines a discrete dynamical system with orbits u_{n} = u(nT), n = 0, 1, 2, …, satisfying the assumptions of §2.1 in a neighborhood D of a hyperbolic fixed point.

In §3 we apply the abstract framework in the context of a system of reactiondiffusion equations discretized in the spatial variables by a standard finite element method, and in the time variable by means of the backward Euler method. The assumptions on L_{h} − L and S_{h} − S are verified by application of rather standard error estimates over the finite time interval [0, T], which we quote from Larsson [11].

Our framework is similar to that of [1] but more flexible. First of all it admits applications with both time and space discretization. In the applications discussed in §3 it also allows us to obtain error bounds of optimal order in both the L_{2} and H^{1} norms. Moreover, it avoids the assumption that S_{h} − 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 Fréchet 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 B_{X}(x, ρ) denotes the closed ball in X with center x and radius ρ.

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

S = L + N (2.1)

in a such a way that, for some constants μ ≥ 1 and κ ∈ (0, 1), the following hypotheses (HL) and (HN) are fulfilled.

Hypothesis (HL). L ∈ L(X), i.e., L is a bounded linear operator in X. Furthermore, X can be decomposed as a direct sum X = X_{1} ⊕ X_{2} of closed subspaces X_{1}, X_{2} that are invariant by L, i.e., LX_{i} ⊂ X_{i}, i = 1, 2. If L_{i} ∈ L(X_{i}), i = 1, 2, denotes the restriction of L to X_{i}, then L_{1} is invertible and

∥L_{1}^{−1}∥_{L(X_{1})} ≤ κ, ∥L_{2}∥_{L(X_{2})} ≤ κ. (2.2)

Moreover, the projections P_{i}, i = 1, 2, associated with the decomposition X = X_{1} ⊕ X_{2} (i.e., P_{i}x = x_{i}, i = 1, 2, for x = x_{1} + x_{2}, x_{i} ∈ X_{i}) satisfy

∥P_{i}∥_{L(X)} ≤ μ. (2.3)

Hypothesis (HN). The mapping N : D → X is Lipschitz continuous with a Lipschitz constant that satisfies

Lip(N) ≤ \frac{1 − κ}{4μ}. (2.4)

Note that the boundedness of the projections P_{1}, P_{2} is a consequence of the closedness of subspaces X_{1}, X_{2}; 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 satisfied and set

σ := \frac{4μ}{1 − κ}. (2.5)

Let i and f be integers, i < f, and let {x̃_{n}}_{n=i}^{f} ⊂ D be a sequence. If {x_{n}}_{n=i}^{f} ⊂ D is an orbit of S, i.e., x_{n+1} = S(x_{n}), n = i, …, f − 1, which satisfies the boundary conditions

P_{2}x_{i} = P_{2}x̃_{i}, P_{1}x_{f} = P_{1}x̃_{f}, (2.6)

then

sup_{i≤n≤f} ∥x_{n} − x̃_{n}∥ ≤ σ sup_{i≤n≤f−1} ∥x̃_{n+1} − S(x̃_{n})∥. (2.7)

(ii) Assume, in addition to (HL), (HN), that the domain D of S contains a closed ball B_{X}(z, ρ) and that

∥z − S(z)∥ ≤ ρ/σ. (2.8)

Then, for any sequence {x̃_{n}}_{n=i}^{f} ⊂ B_{X}(z, ρ/(μσ)), there exists an orbit {x_{n}}_{n=i}^{f} ⊂ B_{X}(z, ρ) of S for which (2.6) and hence (2.7) hold.

Remark 1. With the terminology used in shadowing theory (see, e.g., [6]) we say that {x̃_{n}}_{n=i}^{f} is a δ-pseudo-orbit of S if sup_{i≤n≤f−1} ∥x̃_{n+1} − S(x̃_{n})∥ ≤ δ. If this is the case, then the estimate (2.7) means that {x̃_{n}}_{n=i}^{f} is ϵ-shadowed by the true orbit {x_{n}}_{n=i}^{f} with shadowing distance ϵ ≤ σδ. Part (ii) ensures the existence of a shadow orbit.

Remark 2. Analogous results hold also for infinite sequences {x̃_{n}}_{n=−∞}^{f}, {x̃_{n}}_{n=i}^{∞}, or {x̃_{n}}_{n=−∞}^{∞} with obvious modifications. For instance, for a sequence {x̃_{n}}_{n=−∞}^{f}, the first 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 finite sequences. Note in this connection that the stability constant σ in (2.7) does not depend on the initial and final indices i and f.

The theorem is proved by using the following lemmas.

Lemma 2.2. Assume that L satisfies hypothesis (HL), and that i, f are integers with i < f. Set ν = f − i, X = X^{ν+1}, Y = X_{2} × X^{ν} × X_{1}, and define a linear operator L : X → Y by L : (x_{i}, …, x_{f}) ↦ (y_{i}, …, y_{f+1}), where

y_{i} = P_{2}x_{i}, y_{f+1} = P_{1}x_{f}; y_{n+1} = x_{n+1} − Lx_{n}, n = i, …, f − 1. (2.9)

Then L is invertible and, with respect to the supremum norm of the product spaces X, Y,

∥L^{−1}∥_{L(Y,X)} ≤ \frac{2μ}{1 − κ}. (2.10)

Proof. Given an element y = (y_{i}, …, y_{f+1}) ∈ Y, we define x = (x_{i}, …, x_{f}) ∈ X, by the relations x_{n} = P_{1}x_{n} + P_{2}x_{n}, where

\begin{aligned}P_{1}x_{n} & = (L_{1}^{−1})^{f−n}y_{f+1} − ∑_{j=n+1}^{f}(L_{1}^{−1})^{j−n}P_{1}y_{j}, \\ P_{2}x_{n} & = (L_{2})^{n−i}y_{i} + ∑_{j=i+1}^{n}(L_{2})^{n−j}P_{2}y_{j}. \end{aligned}

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 y_{n}, n = i, …, f + 1, in (2.9) vanish. Then, by (2.9), P_{2}x_{i} = 0 and P_{1}x_{f} = 0. Recursion in (2.9) reveals that P_{2}x_{n} = 0 for n = i + 1, …, f. Similarly, a descending recursion in (2.9) shows that P_{1}x_{n} = 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 definition of x_{n}, n = i, …, f,

∥x_{n}∥ ≤ κ^{f−n}∥y∥ + ∑_{j=n+1}^{f} κ^{j−n}μ∥y∥ + κ^{n−i}∥y∥ + ∑_{j=i+1}^{n} κ^{n−j}μ∥y∥

≤ (1 + ∑_{j=n+1}^{f} κ^{j−n} + κ^{n−i} + ∑_{j=i+1}^{n} κ^{n−j})μ∥y∥

≤ ( ∑_{j=n}^{∞} κ^{j−n} + ∑_{j=−∞}^{n} κ^{n−j})μ∥y∥ = \frac{2μ}{1 − κ}∥y∥.

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, defined in a nonempty subset D of X, is Lipschitz continuous with

α := ∥L^{−1}∥_{L(Y,X)} Lip(N) < 1.

Define S = L + N and set σ := ∥L^{−1}∥_{L(Y,X)}/(1 − α). Then

(2.11) ∥x_{1} − x_{2}∥_{X} ≤ σ∥S(x_{1}) − S(x_{2})∥_{Y}, for all x_{1}, x_{2} ∈ D.

Proof. The bound (2.11) follows readily from the identity

x_{1} − x_{2} = L^{−1}(S(x_{1}) − S(x_{2})) − L^{−1}(N(x_{1}) − N(x_{2})).

Lemma 2.4. In addition to the hypotheses of Lemma 2.3, assume that the domain D of N contains a closed ball B_{X}(z, ρ). Then, for each y ∈ Y with

∥y − S(z)∥_{Y} ≤ ρ/σ, (2.12)

the equation S(x) = y has a unique solution x ∈ B_{X}(z, ρ).

Proof. This is a simple consequence of the contraction mapping theorem. In fact, if we define T(x) = L^{−1}(y − N(x)), then Lip(T) = α < 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 B_{X}(z, ρ) 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^{ν+1} ⊂ X → Y defined by N(x) = y, where

y_{i} = y_{f+1} = 0; y_{n+1} = −N(x_{n}), n = i, …, f − 1. (2.13)

The assumption (HN) implies that Lip(N) ≤ (1 − κ)/(4μ) and (2.10) leads to ∥L^{−1}∥ Lip(N) ≤ 1/2. We can therefore apply Lemma 2.3 with α = 1/2; this yields a value of σ = ∥L^{−1}∥/(1 − α) that coincides with σ in (2.5). Note also that S is defined by S(x) = y, where (see (2.1), (2.9), and (2.13))

y_{i} = P_{2}x_{i}, y_{f+1} = P_{1}x_{f}; y_{n+1} = x_{n+1} − S(x_{n}), 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, ρ) = B(z, ρ)^{ν+1}, where z = (z, …, z). Given {x̃_{n}}_{i=1}^{f} ⊂ B(z, ρ/(μσ)), we put y = (P_{2}x̃_{i}, 0, …, 0, P_{1}x̃_{f}). The condition (2.12) is satisfied. In fact, the first component of y−S(z) is the vector P_{2}x̃_{i}−P_{2}z, whose norm can be estimated by ∥P_{2}∥ ∥x̃_{i}−z∥ ≤ ρ/σ. Similarly, the last component of y − S(z) has norm ≤ ρ/σ. The remaining components of y − S(z) equal 0 − (z − S(z)) and, in view of (2.8), are also bounded in norm by ρ/σ. Since (2.12) is satisfied, the equation S(x) = y has a solution x = (x_{i}, …, x_{f}). The choice of y ensures that {x_{n}}_{n=i}^{f} is the sequence required.

6 STIG LARSSON AND J.-M. SANZ-SERNA

2.2. Shadowing and approximation. We now consider, along with the mapping S in (2.1), a family of approximations {S_{h}}.

Let H be a set of positive numbers with inf H = 0. For each h ∈ H, let X_{h} be a subspace of the Banach space X, in such a way that there exist bounded projections Q_{h} : X → X_{h} and a number γ ≥ 1 with ∥Q_{h}∥_{L(X)} ≤ γ. (2.14) We assume that the spaces X_{h} approximate X in the sense that (2.15) lim_{h→0} Q_{h}u = u, for all u ∈ X. For h ∈ H, we consider mappings S_{h} : D_{h} ⊂ X_{h} → X_{h} with domains D_{h} = D ∩X_{h}, that approximate S in the sense that a continuous positive function ϵ(h) exists such that

lim_{h→0} ϵ(h) = 0 (2.16)

and (2.17) ∥S_{h}(Q_{h}u) − S(u)∥ ≤ ϵ(h), for all u ∈ D such that Q_{h}u ∈ D_{h}. Finally, we assume that S_{h} can be decomposed as S_{h} = L_{h} + N_{h}, (2.18) and that (HL) and (HN) hold for this decomposition. More precisely, this means that L_{h} ∈ L(X_{h}); X_{h} can be decomposed as a direct sum X_{h} = X_{1h} ⊕X_{2h} of closed subspaces invariant by L_{h}; the restrictions L_{ih}, i = 1, 2, of L_{h} to X_{ih} satisfy ∥L_{1h}^{−1}∥_{L(X_{1h})} ≤ κ, ∥L_{2h}∥_{L(X_{2h})} ≤ κ; (2.19) the associated projections satisfy ∥P_{ih}∥_{L(X)} ≤ μ, i = 1, 2; (2.20) and N_{h} : D_{h} → X_{h} with Lipschitz constant

Lip(N_{h}) ≤ \frac{1 − κ}{4μ}.

Note that, X_{ih} is in general different from X_{i} ∩ X_{h}; the latter may well be the trivial subspace {0}. Theorem 2.5. (i) Assume that the subspaces X_{h} of the Banach space X possess the approximation properties (2.14)–(2.15), the mappings S in (2.1) and S_{h} in (2.18) satisfy (HL) and (HN), and the S_{h} approximate S as in (2.17). Let i and f be integers with i < f. Then the following results hold.

(i.a) Let {u_{h,n}}_{n=i}^{f} ⊂ D_{h} be an orbit of S_{h}. If {u_{n}}_{n=i}^{f} is an orbit of S with P_{2}u_{i} = P_{2}u_{h,i}, P_{1}u_{f} = P_{1}u_{h,f}, (2.21) then

sup_{i≤n≤f} ∥u_{n} − u_{h,n}∥ ≤ σϵ(h), (2.22)

where σ is the constant in (2.5).

(i.b) Let {u_{n}}_{n=i}^{f} ⊂ D be an orbit of S with Q_{h}u_{n} ∈ D_{h}, n = i, …, f. (2.23)

If {u_{h,n}}_{n=i}^{f} ⊂ D_{h} is an orbit of S_{h} with

P_{2h}u_{h,i} = P_{2h}Q_{h}u_{i}, P_{1h}u_{h,f} = P_{1h}Q_{h}u_{f}, (2.24)

then

sup_{i≤n≤f} ∥Q_{h}u_{n} − u_{h,n}∥ ≤ γσϵ(h). (2.25)

(ii) Assume that the hypotheses in (i) hold and that S has a fixed point ū such that B_{X}(ū, ρ) ⊂ D for some ρ > 0. Then the following conclusions hold.

(ii.a) For any orbit {u_{h,n}}_{n=i}^{f} ⊂ B_{X}(ū, ρ_{0}), ρ_{0} = ρ/(μσ) < ρ, of S_{h} there is an orbit {u_{n}}_{n=i}^{f} of S for which (2.21) and (2.22) are true.

(ii.b) Let h be small enough for the inequalities

∥ū − Q_{h}ū∥ ≤ ρ/2, 2γσϵ(h) ≤ ρ (2.26)

to hold (cf. (2.15)–(2.16)). Then, for any orbit {u_{n}}_{n=i}^{f} ⊂ B_{X}(ū, ρ_{0}), ρ_{0} = ρ/(2γμσ), of S, the relation (2.23) is satisfied and there is an orbit {u_{h,n}}_{n=i}^{f} of S_{h} for which (2.24) and (2.25) hold.

(ii.c) For h chosen as in (ii.b), the mapping S_{h} has a fixed point that is unique in the ball B_{X_{h}}(Q_{h}ū, ρ/2). Furthermore,

∥Q_{h}ū − ū_{h}∥ ≤ γσϵ(h). (2.27)

Proof. To prove (i.a) we apply part (i) of Theorem 2.1 with x̃_{n} = u_{h,n} ∈ D_{h} ⊂ D, and x_{n} = u_{n}. Then (2.7) yields

sup_{i≤n≤f} ∥u_{n} − u_{h,n}∥ ≤ σ sup_{i≤n≤f−1} ∥u_{h,n+1} − S(u_{h,n})∥.

Since u_{h,n+1} − S(u_{h,n}) = S_{h}(Q_{h}u_{h,n}) − S(u_{h,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 S_{h} playing the role of S, and x̃_{n} = Q_{h}u_{n} ∈ D_{h}, x_{n} = u_{h,n} ∈ D_{h}. The estimate (2.7) reads

sup_{i≤n≤f} ∥u_{h,n} − Q_{h}u_{n}∥ ≤ σ sup_{i≤n≤f−1} ∥Q_{h}u_{n+1} − S_{h}(Q_{h}u_{n})∥,

and (2.25) is a consequence of (2.14), (2.17), in view of the identity

Q_{h}u_{n+1} − S_{h}(Q_{h}u_{n}) = Q_{h}[S(u_{n}) − S_{h}(Q_{h}u_{n})].

Part (ii.a) is a direct consequence of part (ii) of Theorem 2.1 with z = ū.

For (ii.b) we use part (ii) of Theorem (2.1) with S_{h} playing the role of S and z = Q_{h}ū. If v ∈ B_{X_{h}}(Q_{h}ū, ρ/2), then the assumption (2.26) implies

∥v − ū∥ ≤ ∥v − Q_{h}ū∥ + ∥Q_{h}ū − ū∥ ≤ ρ;

this shows that B_{X_{h}}(Q_{h}ū, ρ/2) is contained in B_{X}(ū, ρ), which in turn is assumed to be contained in D. Hence B_{X_{h}}(Q_{h}ū, ρ/2) ⊂ D_{h}. Furthermore, by (2.14), (2.17) and (2.26),

∥Q_{h}ū − S_{h}(Q_{h}ū)∥ = ∥Q_{h}[S(ū) − S_{h}(Q_{h}ū)]∥ ≤ γϵ(h) ≤ \frac{ρ/2}{σ}, (2.28)

so that (2.8) holds with the role of B_{X}(z, ρ) played by B_{X_{h}}(Q_{h}ū, ρ/2). If {u_{n}}_{n=i}^{f} ⊂ B_{X}(ū, ρ_{0}), then, by (2.14), {Q_{h}u_{n}}_{n=i}^{f} ⊂ B_{X_{h}}(Q_{h}ū, ρ/(2μσ)) ⊂ B_{X_{h}}(Q_{h}ū, ρ/2) and (2.23) holds. The existence of an orbit {u_{h,n}}_{n=i}^{f} of S_{h} satisfying (2.24) now follows from Theorem 2.1.

For (ii.c) we apply Lemma 2.4 with X = Y = X_{h}, L = L_{h} − I, N = N_{h}, y = 0 and the role of B_{X}(z, ρ) played by B_{X_{h}}(Q_{h}ū, ρ/2), which we know to be contained in D_{h}. By Lemma 2.8, ∥(L_{h} − I)^{−1}∥ ≤ 2μ/(1 − κ) leading in Lemma 2.3 to α = 1/2 and a value of σ that agrees with the value in (2.5). The condition (2.12) is fulfilled in view of (2.28), because y − S(z) = −S_{h}(Q_{h}ū) + Q_{h}ū. It only remains to prove the bound (2.27). We apply again Lemma 2.4, but this time in the smaller ball B_{X_{h}}(Q_{h}ū, σγϵ(h)) ⊂ B_{X_{h}}(Q_{h}ū, ρ/2); the condition (2.12) is still fulfilled in view (2.28), so that the unique fixed point ū_{h} of S_{h} in B_{X_{h}}(Q)_{h}ū, ρ/2) also lies in B_{X_{h}}(Q_{h}ū, σγϵ(h)).

Our next theorem gives a condition under which the “hyperbolicity” (HL) of the operator L carries over to L_{h}.

Theorem 2.6. Assume that the operator L ∈ L(X), X a Banach space, satisfies hypothesis (HL) and choose κ̃ ∈ (κ, 1), μ̃ > μ. Assume that the subspaces X_{h} with corresponding projections Q_{h} are such that (2.14) holds and that the operators L_{h} ∈ L(X_{h}) approximate L in the sense that

∥L − L_{h}Q_{h}∥_{L(X)} ≤ ϵ(h) (2.29)

with ϵ(h) as in (2.16). Then there exists h_{0} > 0 (depending only on γ, κ, μ, κ̃, μ̃, and the function ϵ) such that, uniformly for h < h_{0}, the operators L_{h} satisfy (HL) with constants κ̃, μ̃.

For the proof of the theorem we need two simple lemmas.

Lemma 2.7. Let A, B ∈ L(X), X a Banach space, with A^{−1} ∈ L(X) and ∥B∥ ≤ \frac{1}{2}∥A^{−1}∥^{−1}. Then (A + B)^{−1} ∈ L(X) and

∥(A + B)^{−1} − A^{−1}∥_{L(X)} ≤ 2∥A^{−1}∥_{L(X)}^{2}∥B∥_{L(X)}.

Proof. We have ∥(A + B)^{−1}∥_{L(X)} = ∥A^{−1} ∑_{n=0}^{∞}(−A^{−1}B)^{n}∥_{L(X)} ≤ 2∥A^{−1}∥_{L(X)} and (A + B)^{−1} − A^{−1} = −(A + B)^{−1}BA^{−1}.

Lemma 2.8. Assume that the operator L ∈ L(X), X a Banach space, satisfies hypothesis (HL) and let ω be a complex number with |ω| = 1. Then (ωI − L)^{−1} ∈ L(X) with

∥(ωI − L)^{−1}∥_{L(X)} ≤ \frac{2μ}{1 − κ}.

Proof. Let |ω| = 1. Assumption (2.2) implies that

(ωI − L_{1})^{−1} = −L_{1} ∑_{n=0}^{∞}(ωL_{1}^{−1})^{n}, (ωI − L_{2})^{−1} = ω^{−1} ∑_{n=0}^{∞}(ω^{−1}L_{2})^{n},

and hence

∥(ωI − L_{1})^{−1}∥_{L(X_{1})} ≤ \frac{κ}{1 − κ}, ∥(ωI − L_{2})^{−1}∥_{L(X_{2})} ≤ \frac{1}{1 − κ}.

Using also (2.3) and (ωI − L)^{−1} = (ωI − L_{1})^{−1}P_{1} + (ωI − L_{2})^{−1}P_{2}, we obtain

∥(ωI − L)^{−1}∥_{L(X)} ≤ \frac{1 + κ}{1 − κ}μ ≤ \frac{2μ}{1 − κ}.

Proof of Theorem 2.6. Define L̃_{h} = L_{h}Q_{h} ∈ L(X) and let |ω| = 1. We begin by showing that (ωI − L̃_{h})^{−1} ∈ L(X). In order to do so we shall apply Lemma 2.7 with A = ωI − L and B = L − L̃_{h}. From Lemma 2.8 we know that ∥(ωI − L)^{−1}∥ ≤ 2μ/(1 − κ). Hence, for h sufficiently small, we have

∥L − L̃_{h}∥_{L(X)} ≤ ϵ(h) ≤ \frac{1 − κ}{4μ} ≤ \frac{1}{2∥(ωI − L)^{−1}∥}.

Now Lemma 2.7 applies and gives (ωI − L̃_{h})^{−1} ∈ L(X) and

∥(ωI − L̃_{h})^{−1} − (ωI − L)^{−1}∥_{L(X)} ≤ 2(\frac{2μ}{1 − κ})^{2}∥L − L̃_{h}∥_{L(X)} ≤ Kϵ(h), (2.30)

where K = 2(2μ/(1 − κ))^{2}.

We next show that (ωI − L_{h})^{−1} ∈ L(X_{h}) and that

(ωI − L_{h})^{−1}Q_{h} = (ωI − L̃_{h})^{−1}Q_{h}. (2.31)

In fact, if f ∈ X, then u = (ωI − L̃_{h})^{−1}Q_{h}f ∈ X satisfies

Q_{h}f = (ωI − L̃_{h})(Q_{h}u + (I − Q_{h})u) = (ωI − L_{h})Q_{h}u + ω(I − Q_{h})u.

We conclude that (I − Q_{h})u = 0, and (ωI − L_{h})Q_{h}u = Q_{h}f, so that u = Q_{h}u = (ωI − L_{h})^{−1}Q_{h}f, which implies (2.31).

We have now proved that, for each sufficiently small h, L_{h} 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 X_{h} = X_{1h} ⊕ X_{2h} as required in assumption (HL). It only remains to prove that the corresponding inequalities (2.19) and (2.20) with constants κ̃, μ̃ hold uniformly with respect to h.

In order to obtain a bound for ∥P_{2h}∥_{L(X_{h})} we first estimate ∥(P_{2} −P_{2h})Q_{h}∥_{L(X)}. Using the representations

P_{2} = \frac{1}{2πi} ∫_{Γ}(ωI − L)^{−1} dω, P_{2h} = \frac{1}{2πi} ∫_{Γ}(ωI − L_{h})^{−1} dω,

where Γ denotes the unit circle with positive orientation, together with (2.31), (2.30), and (2.14) we obtain

∥(P_{2} − P_{2h})Q_{h}∥_{L(X)} = \frac{1}{2π} ∥ ∫_{Γ} ((ωI − L)^{−1} − (ωI − L̃_{h})^{−1}) dω Q_{h}∥_{L(X)} ≤ γKϵ(h).

This implies that, for x ∈ X_{h},

∥P_{2h}x∥ ≤ ∥P_{2}x∥ + ∥(P_{2} − P_{2h})x∥ ≤ (μ + γKϵ(h))∥x∥,

so that

∥P_{2h}∥_{L(X_{h})} ≤ μ + γKϵ(h) ≤ μ̃,

provided that h is sufficiently small. Since (P_{1} − P_{1h})Q_{h} = (P_{2h} − P_{2})Q_{h}, we also have ∥P_{1h}∥_{L(X_{h})} ≤ μ̃.

We now turn to the bound for ∥L_{2h}∥_{L(X_{2h})}. Since

L_{2}P_{2} = \frac{1}{2πi} ∫_{Γ} ω(ωI − L)^{−1} dω, L_{2h}P_{2h} = \frac{1}{2πi} ∫_{Γ} ω(ωI − L_{h})^{−1} dω,

we have

\begin{aligned}∥(L_{2}P_{2} − L_{2h}P_{2h})Q_{h}∥_{L(X)} & = \frac{1}{2π} ∥ ∫_{Γ} ω((ωI − L)^{−1} − (ωI − L̃_{h})^{−1}) dω Q_{h}∥_{L(X)} \\ & ≤ γKϵ(h). \end{aligned}

Hence, for x ∈ X_{2h},

\begin{aligned}∥L_{2h}x∥ & = ∥L_{2h}P_{2h}x∥ ≤ ∥(L_{2h}P_{2h} − L_{2}P_{2})Q_{h}x∥ + ∥L_{2}P_{2}x∥ \\ & ≤ γKϵ(h)∥x∥ + κ∥P_{2}x∥. \end{aligned}

Here, since Q_{h}x = P_{2h}x = x,

∥P_{2}x∥ ≤ ∥(P_{2} − P_{2h})Q_{h}x∥ + ∥P_{2h}x∥ ≤ γKϵ(h)∥x∥ + ∥x∥,

so that

∥L_{2h}x∥ ≤ (κ + 2γKϵ(h))∥x∥,

and we conclude that, for small h,

∥L_{2h}∥_{L(X_{2h})} ≤ κ + 2γKϵ(h) ≤ κ̃.

The required bound for ∥L_{1h}^{−1}∥_{L(X_{1h})} is obtained in the same way, using the representations

L^{−1}P_{1} = −\frac{1}{2πi} ∫_{Γ} ω^{−1}(ωI − L)^{−1} dω, L_{1h}^{−1}P_{1h} = −\frac{1}{2πi} ∫_{Γ} ω^{−1}(ωI − L_{h})^{−1} dω.

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 finite element approximation of a system of reaction-diffusion equations.

3.1. The continuous problem. We consider the model problem

\begin{aligned}u_{t} − DΔu & = f̃(u), & x ∈ Ω, t > 0, \\ u & = 0, & x ∈ ∂Ω, t > 0, \\ u(·, 0) & = u_{0}, & x ∈ Ω, \end{aligned} (3.1)

where Ω is a bounded domain in R^{d}, d = 1, 2, 3, u = u(x, t) ∈ R^{s}, u_{t} = ∂u/∂t, Δu = ∑_{i} ∂^{2}u/∂x_{i}^{2}, D = diag(d_{1}, …, d_{s}) is a diagonal matrix of constant coefficients d_{i} > 0, and f̃: R^{s} → R^{s} is continuously differentiable. We assume that Ω is either a convex polygon or has a smooth boundary. If d = 2, 3 we assume, in addition, that the Jacobian of f̃ satisfies the growth condition

|f̃′(ξ)| ≤ C(1 + |ξ|^{δ}), ξ ∈ R^{s},

where | · | denotes the Euclidean norm on R^{s} and the induced matrix norm, and where δ = 2 if d = 3, δ ∈ [0, ∞) if d = 2.

In the sequel we use the Hilbert space H = (L_{2}(Ω))^{s}, with its standard norm ∥ · ∥ and inner product (·, ·). The norms in the Sobolev spaces (H^{m}(Ω))^{s}, m ≥ 0 are denoted by ∥ · ∥_{m}. The space V = (H_{0}^{1}(Ω))^{s}, with norm ∥ · ∥_{1}, consists of the functions in (H^{1}(Ω))^{s} that vanish on ∂Ω. We define the operator A = −DΔ with domain D(A) = (H^{2}(Ω) ∩ H_{0}^{1}(Ω))^{s}. Then A is a closed, densely defined and selfadjoint operator in H with compact inverse. Moreover, our assumptions guarantee that the mapping f̃ induces an operator f : V → H through f(v)(x) = f̃(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:

u′ + Au = f(u), t > 0; u(0) = u_{0}. (3.2)

We assume further that (3.2) has a stationary solution ū with ū ∈ D(A), Aū = f(ū); by standard embedding results ū is continuous in the closure of Ω. The formula (Bv)(x) = f̃′(ū(x))v(x) clearly defines an operator B ∈ L(H). The operator A = A − B is the linearization of A − f at ū 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 ū is “hyperbolic”, i.e., that the spectrum of A does not intersect the imaginary axis. Let P_{1} and P_{2} the projections respectively associated with the sets σ_{1} = σ(A) ∩ {Re z < 0} and σ_{2} = σ(A) ∩ {Re z > 0} that partition the spectrum σ(A) of A and let H_{1} and H_{2} be the ranges of P_{1} and P_{2}. It follows that H is a direct sum H = H_{1} ⊕ H_{2}; the subspaces H_{i} are invariant under A and, if A_{i}, i = 1, 2, denotes the restriction of A to H_{i}, then A_{1} ∈ L(H_{1}), D(A_{2}) = D(A) ∩ H_{2}. Furthermore, there are M ≥ 1 and α > 0, such that,

\begin{aligned}∥e^{−tA_{1}}v∥_{m} & ≤ Me^{αt}∥v∥, & t ≤ 0, v ∈ H_{1}, m = 1, 2, \\ ∥e^{−tA_{2}}v∥_{m} & ≤ Mt^{−m/2}e^{−αt}∥v∥, & t > 0, v ∈ H_{2}, m = 1, 2, \\ ∥e^{−tA_{2}}v∥_{1} & ≤ Me^{−αt}∥v∥_{1}, & t ≥ 0, v ∈ H_{2} ∩ V. \end{aligned} (3.3)

We refer to [9, §1.5] for these facts.

Since H_{1} ⊂ D(A), we see that we also have a direct sum V = V_{1} ⊕ V_{2}, where V_{1} = H_{1} and V_{2} = H_{2} ∩ V, with associated projections P_{1}|_{V} and P_{2}|_{V}. By the closed graph theorem, we may select a constant μ ≥ 1 such that

∥P_{i}∥_{L(H)} ≤ μ, ∥P_{i}∥_{L(V)} ≤ μ, i = 1, 2. (3.4)

By combination of these with (3.3) we have

\begin{aligned}∥e^{−tA}v∥_{1} & ≤ Ct^{−1/2}e^{αt}∥v∥, & t > 0, v ∈ H, \\ ∥e^{−tA}v∥_{1} & ≤ Ce^{αt}∥v∥_{1}, & t ≥ 0, v ∈ V. \end{aligned} (3.5)

With F (v) = f(v) − Bv, we may rewrite (3.2) as

u′ + Au = F (u), t > 0; u(0) = u_{0}, (3.6)

and clearly we also have

Aū = F (ū). (3.7)

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 sufficiently small neighborhood of ū.

Lemma 3.1. If v, w ∈ B_{V} (ū, ρ), then

∥F (v) − F (w)∥ ≤ k(ρ)∥v − w∥_{1}, (3.8)

where k(ρ) = O(ρ) as ρ → 0.

The initial value problem (3.6) (or (3.2)) has a unique local solution for any initial datum u_{0} ∈ V, see [9, Theorem 3.3.3]. We denote by S(t, ·) the corresponding (local) solution operator, so that u(t) = S(t, u_{0}) is the solution of (3.6). The following lemma shows that the local solutions can be extended in time, if they start sufficiently near ū.

Lemma 3.2. For each ρ_{1} > 0 and T > 0 there is ρ > 0 such that, if u_{0} ∈ B_{V} (ū, ρ), then S(t, u_{0}) is defined and belongs to B_{V} (ū, ρ_{1}) for t ∈ [0, T].

Proof. Let ρ_{1}, T > 0 be given. For ρ > 0 let τ ∈ [0, T] be the largest time such that u_{0} ∈ B_{V} (ū, ρ) implies that u(t) = S(t, u_{0}) exists and belongs to B_{V} (ū, ρ_{1} + 1) for t ∈ [0, τ]. We must choose ρ such that τ = T.

Let z(t) = u(t) − ū. Forming the difference between (3.6) and (3.7) and using the variation of constants formula, we obtain

z(t) = e^{−tA}z(0) + ∫_{0}^{t} e^{−(t−s)A}(F (u(s)) − F (ū)) ds.

Invoking (3.5) and (3.8), we therefore have, for t ∈ [0, τ],

\begin{aligned}∥z(t)∥_{1} & ≤ Ce^{αt}∥z(0)∥_{1} + C ∫_{0}^{t}(t − s)^{−1/2}e^{α(t−s)}∥F (u(s)) − F (ū)∥ ds \\ & ≤ Ce^{αT} (ρ + k(ρ_{1} + 1) ∫_{0}^{t}(t − s)^{−1/2}∥z(s)∥_{1} ds). \end{aligned}

Gronwall’s lemma (see [13, Lemma 5.6.7] or [9, Exercise 4 of §6.1]) now yields

∥z(t)∥_{1} ≤ C(ρ_{1}, T)ρ, t ∈ [0, τ],

so that if we choose ρ = ρ_{1}/C(ρ_{1}, T), then

∥z(t)∥_{1} ≤ ρ_{1}, t ∈ [0, τ].

If τ < T, then by local existence we obtain a contradiction with the maximality of τ. Hence, S(t, u_{0}) is defined and belongs to B_{V} (ū, ρ_{1}) for t ∈ [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 ρ_{1} > 0, T > 0 and assume that u(t) = S(t, u_{0}) exists and belongs to B_{V} (ū, ρ_{1}) for t ∈ [0, T]. Then there exists C(ρ_{1}, T) such that

∥u(t)∥_{2} ≤ C(ρ_{1}, T) t^{−1/2}, t ∈ (0, T].

Proof. In view of (3.8) we have ∥F (u(t))∥ ≤ C(ρ_{1}) for t ∈ [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 §2, we choose T such that

κ := Me^{−αT} < 1, (3.9)

where M and α are the constants in (3.3). We then define S = S(T, ·), L = e^{−T A}, N = S − L. It is clear from the above that assumption (HL) is satisfied, 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 ϵ > 0 there is ρ > 0 such that v_{1}, v_{2} ∈ B_{V} (ū, ρ) implies

∥N(v_{1}) − N(v_{2})∥_{m} ≤ ϵ∥v_{1} − v_{2}∥_{m}, m = 0, 1.

Proof. Let T be as in (3.9) and let ρ_{1} > 0. We first carry out an a priori estimation under the assumption that u_{i} = S(t, v_{i}), i = 1, 2, exist and belong to B_{V} (ū, ρ_{1}) for t ∈ [0, T]. From the variation of constants formula

u_{i}(t) = e^{−tA}v_{i} + ∫_{0}^{t} e^{−(t−s)A}F (u_{i}(s)) ds, t ∈ [0, T],

so that N(v_{i}) = w_{i}(T), where

w_{i}(t) = ∫_{0}^{t} e^{−(t−s)A}F (u_{i}(s)) ds.

Using (3.5) and (3.8), we obtain

\begin{aligned}∥w_{1}(t) − w_{2}(t)∥_{1} & ≤ Ce^{αt} ∫_{0}^{t}(t − s)^{−1/2}∥F (u_{1}(s)) − F (u_{2}(s))∥ ds \\ & ≤ C(T)k(ρ_{1}) ∫_{0}^{t}(t − s)^{−1/2}∥u_{1}(s) − u_{2}(s)∥_{1} ds. \end{aligned}

Here u_{1}(s) − u_{2}(s) = e^{−sA}(v_{1} − v_{2}) + w_{1}(s) − w_{2}(s), so that another application of (3.5) yields, for t ∈ [0, T],

\begin{aligned}∥w_{1}(t) − w_{2}(t)∥_{1} & ≤ C(T)k(ρ_{1})∥v_{1} − v_{2}∥ \\ & + C(T, ρ_{1}) ∫_{0}^{t}(t − s)^{−1/2}∥w_{1}(s) − w_{2}(s)∥_{1} ds. \end{aligned}

Gronwall’s lemma now shows that

∥w_{1}(t) − w_{2}(t)∥_{1} ≤ C(T, ρ_{1})k(ρ_{1})∥v_{1} − v_{2}∥, t ∈ [0, T]. (3.10)

This is the required a priori bound and we may now complete the proof. Let ϵ > 0. Since k(ρ_{1}) = O(ρ_{1}) and C(T, ρ_{1}) = O(1) as ρ_{1} → 0, we may choose ρ_{1} so that C(T, ρ_{1})k(ρ_{1}) ≤ ϵ. Lemma 3.2 provides ρ such that v_{1}, v_{2} ∈ B_{V} (ū, ρ) implies u_{1}(t), u_{2}(t) ∈ B_{V} (ū, ρ_{1}) for t ∈ [0, T], and (3.10) then yields

∥w_{1}(T) − w_{2}(T)∥_{1} ≤ ϵ∥v_{1} − v_{2}∥,

which implies both the required estimates.

Lemma 3.4 shows that there is ρ such that, if we set D = B_{V} (ū, ρ), then N satisfies (HN) with both X = H and X = V. Moreover, we have found a larger radius ρ_{1} > ρ such that

u_{0} ∈ D = B_{V} (ū, ρ) ⇒ S(t, u_{0}) ∈ B_{V} (ū, ρ_{1}), t ∈ [0, T]. (3.11)

In summary, we have chosen the parameters so as to make sure that S = L + N satisfies (HL) and (HN) in both X = H and X = V.

Remark 3. Note that L is the linearization of S at ū. In fact, the mapping S : D ⊂ V → V is Fréchet differentiable with L = S′(ū) ∈ L(V). However, the mapping S : D ⊂ H → H is not differentiable, because D = B_{V} (ū, ρ) is not a neighborhood of ū with respect to the topology of H.

3.2. The discrete problem. In this section we first discretize the initial-boundary value problem (3.1) with respect to the spatial variables by means of a standard piecewise linear finite element method and apply the shadowing results of §2.2. At the end of the section we then briefly discuss completely discrete approximations obtained by means of the backward Euler time-stepping.

Let {V_{h}}_{0<h<1} be a family of finite dimensional subspaces of V, where each V_{h} consists of continuous piecewise polynomials of degree ≤ 1 with respect to a triangulation of Ω with maximal mesh size h, see [5]. The approximate solution u_{h}(t) ∈ V_{h} of (3.1) is defined by

\begin{aligned}(u′_{h}, χ) + (D∇u_{h}, ∇χ) & = (f̃(u_{h}), χ), ∀χ ∈ V_{h}, t > 0, \\ u_{h}(0) & = u_{0h} \end{aligned} , (3.12)

where = u_{0h} ∈ V_{h} is an approximation of u_{0}.

We want to set this problem in the framework of §2.2 with X_{h} = V_{h}, and both X = H and X = V. Let Q_{h} : H → V_{h} be the orthogonal projection. Then Q_{h} satisfies (2.14) (with δ = 1) if X = H. In order to satisfy (2.14) with X = V we assume that Q_{h} is bounded (uniformly in h) with respect to the H^{1} norm. It is easy to see that this is true if the spaces V_{h} satisfy an inverse assumption. For a more general discussion of the H^{1} boundedness of Q_{h} we refer to [7].

It is well known that (in view of standard interpolation error bounds and the L_{2} and H^{1} boundedness of Q_{h})

∥Q_{h}v − v∥_{m} ≤ Ch^{2−m}∥v∥_{2}, v ∈ D(A), m = 0, 1. (3.13)

Since D(A) is dense in H it follows that (2.15) holds with X = H and X = V. Introducing the linear operator A_{h} : V_{h} → V_{h} defined by

(A_{h}ψ, χ) = (D∇ψ, ∇χ), ∀ψ, χ ∈ V_{h},

and with f : V → H defined as before, we may write (3.12) as

u′_{h} + A_{h}u_{h} = Q_{h}f(u_{h}), t > 0; u_{h}(0) = u_{0h}, (3.14)

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 S_{h}(t, ·), such that u_{h}(t) = S_{h}(t, u_{0h}) 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(−tA_{h}), and the local Lipschitz condition for the mapping f : V → H, see [11].

With A_{h} = A_{h} − Q_{h}B and F (v) = f(v) − Bv as before, we rewrite (3.14) as

u′_{h} + A_{h}u_{h} = Q_{h}F (u_{h}), t > 0; u_{h}(0) = u_{0h},

which is the discrete version (3.6). Since A_{h} is selfadjoint, positive definite (uniformly in h), and Q_{h}B is bounded, we deduce that A_{h} is sectorial (uniformly in h), so that for some c > 0

\begin{aligned}∥e^{−tA_{h}}v∥_{1} & ≤ Ce^{ct}∥v∥_{1}, & t ≥ 0, v ∈ V_{h}, \\ ∥e^{−tA_{h}}v∥_{1} & ≤ Ct^{−1/2}e^{ct}∥v∥, & t > 0, v ∈ V_{h}, \end{aligned} (3.15)

which are discrete versions of (3.5). Here we have employed the equivalence of norms ∥v∥_{1} ≈ ∥A_{h}^{1/2}v∥ for v ∈ V_{h}. The inequalities (3.15) can also be proved by noting that u_{h}(t) = e^{−tA_{h}}u_{0h} satisfies (3.14) with f(u_{h}) replaced by Bu_{h}, 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 < τ ≤ T and assume that S(t, u_{0}), S_{h}(t, u_{0h}) ∈ B_{V} (ū, ρ) for t ∈ [0, τ]. Then, for t ∈ [0, τ], we have

\begin{aligned}∥S_{h}(t, u_{0h}) − S(t, u_{0})∥ & ≤ C(ρ, T)(∥u_{0h} − Q_{h}u_{0}∥ + h^{2}t^{−1/2}), \\ ∥S_{h}(t, u_{0h}) − S(t, u_{0})∥_{1} & ≤ C(ρ, T)t^{−1/2}(∥u_{0h} − u_{0}∥ + h). \end{aligned}

The following is a discrete analogue of Lemma 3.2.

Lemma 3.6. For each ρ_{1} > 0, T > 0 there are ρ > 0, h_{0} > 0 such that, if u_{0h} ∈ B_{V} (ū, ρ)∩V_{h} and h < h_{0}, then S_{h}(t, u_{0h}) exists and belongs to B_{V} (ū, ρ_{1})∩V_{h} for t ∈ [0, T].

Proof. Let ρ_{1}, T > 0 be given. For ρ > 0 let τ ∈ [0, T] be the largest time such that u_{0h} ∈ B_{V} (ū, ρ)∩V_{h} implies that S_{h}(t, u_{0h}) exists and belongs to B_{V} (ū, ρ_{1}+1)∩V_{h} for t ∈ [0, τ]. By local existence there are t_{0} > 0 and ρ > 0 such that u_{0h} ∈ B_{V} (ū, ρ)∩V_{h} implies that S_{h}(t, u_{0h}) exists and belongs to B_{V} (ū, ρ_{1})∩V_{h} for t ∈ [0, t_{0}]. Moreover, the second error estimate of Lemma 3.5 gives the a priori estimate

\begin{aligned}∥S_{h}(t, u_{0h}) − ū∥_{1} & ≤ C(ρ_{1}, T)t^{−1/2}(∥u_{0h} − ū∥ + h) \\ & ≤ C(ρ_{1}, T)t_{0}^{−1/2}(ρ + h_{0}) ≤ ρ_{1}, t ∈ [t_{0}, τ], \end{aligned}

provided that ρ and h_{0} are sufficiently small. If τ < T, then by local existence we obtain a contradiction with the maximality of τ. Hence, S_{h}(t, u_{0}) is defined and belongs to B_{V} (ū, ρ_{1}) ∩ V_{h} for t ∈ [0, T].

We will also use the error bounds

\begin{aligned}∥e^{−tA_{h}}Q_{h}v − e^{−tA}v∥ & ≤ Ch^{2}t^{−1}e^{αt}∥v∥, & t > 0, v ∈ H, \\ ∥e^{−tA_{h}}Q_{h}v − e^{−tA}v∥_{1} & ≤ Cht^{−1/2}e^{αt}∥v∥_{1}, & t > 0, v ∈ V, \end{aligned} (3.16)

which can be proved by the using the techniques of [11].

With T as in (3.9) we define L_{h} = e^{−T A_{h}}, and (3.16) shows

∥L_{h}Q_{h} − L∥_{L(H)} + h∥L_{h}Q_{h} − L∥_{L(V)} ≤ C(T)h^{2}.

The assumption (2.29) is thus satisfied with both X = H and X = V. We conclude that Theorem 2.6 applies, showing that, for small h, L_{h} satisfies (HL) with slightly larger constants κ̃ > κ, μ̃ > μ. Adjusting κ, μ, we may conclude that (2.19), (2.20) hold.

Finally, we define S_{h} = S_{h}(T, ·), N_{h} = S_{h} − L_{h}, 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 ρ, h_{0} such that, for h < h_{0}, N_{h} satisfies (HN) with D_{h} = D ∩ V_{h}, D = B_{V} (ū, ρ). The argument also selects ρ_{1} such that, in analogy with (3.11),

u_{0h} ∈ D_{h} ⇒ S_{h}(t, u_{0h}) ∈ B_{V} (ū, ρ_{1}), t ∈ [0, T]. (3.17)

Moreover, using Lemma 3.5 together with (3.11) and (3.17), we see that, if v ∈ D, Q_{h}v ∈ D_{h}, then

∥S_{h}(Q_{h}v) − S(v)∥ ≤ C(ρ_{1}, T)h^{2}, ∥S_{h}(Q_{h}v) − S(v)∥_{1} ≤ C(ρ_{1}, T)h,

since ∥Q_{h}v − v∥ ≤ Ch∥v∥_{1} ≤ Chρ. We conclude that (2.17) holds with X = H and ϵ(h) = Ch^{2}, and with X = V and ϵ(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 ρ_{0}, h_{0}, and C such that, for any h < h_{0}, the following hold:

(1) If u_{h} is a solution of (3.12) with u_{h}(t) ∈ B_{V} (ū, ρ_{0}) for t ∈ [0, T ], then there is a solution u of (3.1) such that

∥u_{h}(t) − u(t)∥_{m} ≤ C(1 + t^{−1/2})h^{2−m}, t ∈ (0, T ], m = 0, 1. (3.18)

(2) Conversely, if u is a solution of (3.1) with u(t) ∈ B_{V} (ū, ρ_{0}) for t ∈ [0, T ], then there is a solution u_{h} of (3.12) such that (3.18) holds.

(3) Equation (3.12) has a stationary solution ū_{h} such that

∥ū_{h} − ū∥_{1} ≤ Ch.

Proof. Let ρ, ρ_{1}, h_{0}, T be as above. Choose ρ_{0} and adjust h_{0} in such a way that the requirements of parts (ii.a) and (ii.b) of Theorem 2.5 are satisfied with X = V.

(1) Let u_{h}(t) ∈ B_{V} (ū, ρ_{0}) for t ∈ [0, T ] and apply part (ii.a) with X = V to the sequence u_{h}(nT), nT ∈ [0, T ], which is an orbit of S_{h}. This gives the existence of an orbit u(nT), nT ∈ [0, T ], of S, satisfying (2.21) of part (i.a), and hence (2.22) gives the special case m = 1 of the inequality

∥u_{h}(nT) − u(nT)∥_{m} ≤ Ch^{2−m}, nT ∈ [0, T ], m = 0, 1. (3.19)

Another application of part (i.a), now with X = H, proves the case m = 0 of (3.19).

From the sequence u(nT) we define u(t) = S(t−nT, u(nT)) for t ∈ [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 ∈ [0, T] we have

∥u_{h}(t) − u(t)∥_{1} ≤ C(ρ_{1}, T)t^{−1/2}(∥u_{h}(0) − u(0)∥ + h) ≤ C(ρ_{1}, T)t^{−1/2}h.

For t ∈ [(n + 1)T, (n + 2)T], n ≥ 0, we have

∥u_{h}(t) − u(t)∥_{1} ≤ C(ρ_{1}, 2T)t^{−1/2}(∥u_{h}(nT) − u(nT)∥ + h) ≤ C(ρ_{1}, T)h.

This proves the special case m = 1 of (3.18). The case m = 0 is obtained similarly.

(2) Let u(t) ∈ B_{V} (ū, ρ_{0}) for t ∈ [0, T ] and apply part (ii.b) with X = V to the sequence u(nT), nT ∈ [0, T ], which is an orbit of S. This gives the existence of an orbit u_{h}(nT), nT ∈ [0, T ], of S_{h}, satisfying (2.24) of part (i.b), and hence (2.25) gives the special case m = 1 of the inequality

∥Q_{h}u(nT) − u_{h}(nT)∥_{m} ≤ Ch^{2−m}, nT ∈ [0, T ], m = 0, 1. (3.20)

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,

∥Q_{h}u(t) − u(t)∥_{m} ≤ Ch^{2−m}∥u(t)∥_{2} ≤ C(ρ_{1}, T)h^{2−m}t^{−1/2}.

(3) Part (ii.c) of Theorem 2.5 gives ū_{h} and the error bound is an immediate consequence of (2.27) and (3.13).

We conclude this section by briefly 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

(U_{j} − U_{j−1})/k + A_{h}U_{j} = Q_{h}f(U_{j}), t_{j} = jk > 0; U_{0} = u_{0h}.

The local solution operator S_{h,k}(t_{j}, u_{0h}) is readily obtained by using the smoothing property of the corresponding linear evolution operator E_{h,k}(t_{j}) = (I − kA_{h})^{−j}, and the local Lipschitz condition for f : V → H, see [11]. This smoothing property carries over to the linearized operator E_{h,k}(t_{j}) = (I − kA_{h})^{−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 E_{h,k}(t_{j}) analogous to those in (3.16) may be found in [11], and with a discrete time T suitably chosen we find that L_{h,k} = E_{h,k}(T) satisfies (HL). Setting S_{h,k} = S_{h,k}(T, ·), N_{h,k} = S_{h,k} −L_{h,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

∥U_{j} − u(t_{j})∥_{m} ≤ C((1 + t_{j}^{−1/2})h^{2−m} + (1 + t_{j}^{−(m+1)/2})k), t_{j} ∈ (0, T ], m = 0, 1.

Remark 4. The framework of §2 applies also in the context of a finite element method for the Cahn-Hilliard equation, for which the finite time analysis was carried out in [8]. We skip the details.

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Department of Mathematics, Chalmers University of Technology and Göteborg University, S–412 96 Göteborg, Sweden

E-mail address: stig@math.chalmers.se

Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

E-mail address: sanzserna@cpd.uva.es