NUMERICAL SHADOWING NEAR THE GLOBAL ATTRACTOR FOR A SEMILINEAR PARABOLIC EQUATION

STIG LARSSON AND SERGEI YU. PILYUGIN

Abstract. We use the shadowing approach to study the long-time behavior of numerical approximations of semilinear parabolic equations. We show that the corresponding nonlinear semigroup has a Lipschitz shadowing property in a neighborhood of its global attractor. The proof is based on reduction to an inertial manifold and application of shadowing techniques developed for finitedimensional systems. When applied to a semilinear parabolic problem in one space variable, approximated by a standard finite element method in space and by backward Euler time-stepping, our result yields, for any computed trajectory near the attractor, an exact shadow trajectory with an optimal error bound uniformly in time.

1. Introduction

In this paper we use the shadowing approach to study the long-time behavior of numerical approximations (discretized both in space and time) of semilinear parabolic equations.

Let S(t) be the nonlinear semigroup defined by the partial differential equation

\begin{matrix}u_{t} − u_{xx} = f(u), & 0 < x < 1, t > 0, \\ u(0, t) = u(1, t) = 0, & t > 0, \\ u(x, 0) = u_{0}(x), & 0 < x < 1, \end{matrix} (1)

so that u(x, t) = (S(t)u_{0})(x), and let U_{n}(x) = (S_{hk}(t_{n})u_{0})(x) denote the numerical approximation obtained by a standard finite element method in the spatial variable and backward Euler time stepping. Our goal is to estimate the differences between exact solutions and numerical approximations uniformly over infinitely long time intervals. For this purpose we prove that the discrete semigroup S(1) has a Lipschitz shadowing property in a neighborhood of its global attractor A. The proof of this result is accomplished by first reducing the shadowing problem to its analogue on an inertial manifold containing A, and then applying shadowing techniques developed for finite-dimensional structurally stable systems [19].

Roughly speaking, when applied to the semilinear parabolic problem (1), our shadowing result implies that, given a computed trajectory U_{n} = S_{hk}(t_{n})u_{0}, we can find a time t_{N} and an initial value u_{N} such that u(t_{n}) = S(t_{n} − t_{N})u_{N} is close to U_{n} = S_{hk}(t_{n})u_{0} for t_{n} ≥ t_{N}, with an error bound of optimal order uniformly in time.

Date: April 22, 1998.

1991 Mathematics Subject Classification. 65M15, 65M60, 35B40.

Key words and phrases. semilinear, parabolic, Lipschitz shadowing property, attractor, Morse-Smale, inertial manifold, numerical method, finite element, backward Euler, error estimate, longtime.

Partially supported by the Swedish Research Council for Engineering Sciences (TFR) and by INTAS grant 96–1158.

Partially supported by INTAS grant 96–1158.

In earlier work, see, e.g., Beyn [4], [5], Alouges and Debussche [1], [2], Larsson and Sanz-Serna [14], [15], similar results were obtained, but only in a neighborhood of a part of the attractor, such as a hyperbolic fixed point, periodic orbit, or connecting orbit.

In order to prove the Lipschitz shadowing property in a whole neighborhood of the attractor A we rely heavily on the assumption that S has Morse-Smale structure on A. This is the main reason why we restrict our attention to the semilinear parabolic problem (1) in one space variable, for which this is known to be generically true; see below. Hale and Raugel [10] proved perturbation results for the whole attractor (also based on the Morse-Smale structure), but their error bounds are of lower order of convergence. However, although our present result gives optimal order error bounds for shadowing trajectories, it does not immediately lead to an optimal order bound for the distance between the attractors of S and S_{hk}. This problem will be addressed in future work.

2. The Shadowing Problem

We consider the initial-boundary value problem (1). We use the standard Hilbert space L_{2} = L_{2}(0, 1) with inner product and norm

(v, w) = ∫_{0}^{1} v(x)w(x) dx, ∥v∥ = ( ∫_{0}^{1} |v(x)|^{2} dx)^{1/2}, (2)

and the standard Sobolev spaces H^{l} = H^{l}(0, 1), l = 1, 2, with norms

∥v∥_{l} = ( ∑_{j=0}^{l} ∥v^{(j)}∥^{2})^{1/2}, (3)

and H_{0}^{1} = {v ∈ H^{1} : v(0) = v(1) = 0}. Below we denote H_{0}^{1} by H, we write |v| instead of ∥v∥_{1}, and dist(Y, X) is the distance between Y, X ⊂ H.

Defining the operator A = −D_{x}^{2} with domain D(A) = H ∩ H^{2} we write (1) as an initial value problem in H,

u̇ + Au = F (u), t > 0; u(0) = u_{0}, (4)

and denote its solution by u(t) = S(t)u_{0}. We assume that f belongs to C^{2}. Then the induced mapping F : H → L_{2} satisfies a local Lipschitz condition: for any bounded set B ⊂ H there is a constant C(B) such that

∥F (v) − F (w)∥ ≤ C(B)|v − w|, for v, w ∈ B.

A standard argument based on writing (4) as fixed point equation,

u(t) = e^{−tA}u_{0} + ∫_{0}^{t} e^{−(t−s)A}F (u(s)) ds,

now gives local solvability: for any bounded set B_{0} ⊂ H there is t^{∗} = t^{∗}(B_{0}) > 0 and a bounded set B ⊂ H such that for all u_{0} ∈ B_{0} (4) has a unique solution u(t) = S(t)u_{0} ∈ B for t ∈ [0, t^{∗}].

We further assume that F is such that solutions of (4) satisfy a global a priori bound: for any bounded set B_{0} ⊂ H there is a bounded set B ⊂ H such that

S(t)B_{0} ⊂ B, t ∈ [0, ∞). (5)

For example, it is sufficient to assume that there is a constant C such that

sf(s) ≤ C, for s ∈ R. (6)

Then local solutions are extended globally in t and, hence, S(t) is a semigroup of nonlinear operators on H. It is also known that under our conditions S(t)u_{0} depends smoothly on u_{0} and that for a fixed T > 0 and for any bounded set B ⊂ H there exists C(T, B) such that

|S(t)v − S(t)w| ≤ C(T, B)|v − w|, for v, w ∈ B, t ∈ [0, T]. (7)

We next describe two Properties a and b that are crucial to our analysis.

Property a. S(t) has a global attractor A in H, i.e., there is a compact subset A of H with the following properties: a1. A is invariant, i.e., S(t)A = A for t ∈ R; a2. A is uniformly globally attractive, i.e., for any ϵ > 0 and for any bounded set

B ⊂ H there exists T > 0 such that

dist(S(t)B, A) < ϵ for t ≥ T.

a3. A is Lyapunov stable, i.e., for any neighborhood W of A there is a neighborhood U of A such that S(t)U ⊂ W for t ∈ [0, ∞).

Note that a3 is a consequence of a2.

The existence of a global attractor A implies the existence of a bounded absorbing set H_{0}, such that for each bounded set B_{0} ⊂ H there is T such that

S(t)B_{0} ⊂ H_{0} for t ∈ [T, ∞).

Condition (6) implies [12] that S(t) has a global attractor.

The second Property b is that S(t) has Morse-Smale structure on A, as described below. First we give some definitions.

Note that a fixed point p = p(x) of S(t) is a solution of the boundary value problem:

p_{xx} + f(p) = 0, p(0) = p(1) = 0. (8)

As usual, we say that a fixed point p is a hyperbolic fixed point of S(t) if the spectrum of the derivative DS(t)(p) does not intersect the unit circle [11]. In the present situation one can reformulate this condition as follows. A fixed point p = p(x) of S(t) is hyperbolic if and only if 0 is not an eigenvalue of the linear variational operator

v ↦ v_{xx} + f′(p(x))v

with the Dirichlet boundary conditions. In this case we say that p(x) is a hyperbolic solution of (8).

If p is hyperbolic, then the stable manifold of p with respect to S(t), defined by

W_{S}^{s}(p) = {u ∈ H : S(t)u → p as t → ∞},

and the unstable manifold of p, defined by

W_{S}^{u}(p) = {u ∈ H : S(t)u exists for t ≤ 0 and S(t)u → p as t → −∞},

are smooth immersed submanifolds of H [11]. If p, q are hyperbolic fixed points of S(t), then we say that W_{S}^{u}(p) and W_{S}^{u}(q) are transversal if for any v ∈ W_{S}^{u}(p) ∩ W_{S}^{s}(q) the direct sum of the tangent spaces T_{v}W_{S}^{u}(p) and T_{v}W_{S}^{s}(q) equals H [12].

Property b. S(t) has Morse-Smale structure on A, i.e., b1. S(t) has a finite number of fixed points π_{1}, …, π_{N}, and these points are

hyperbolic; b2. every trajectory of S(t) tends to a fixed point, i.e.,

H = ⋃_{1≤i≤N} W_{S}^{s}(π_{i})

b3. the attractor A is the union of the unstable manifolds of the fixed points of S(t), i.e.,

A = ⋃_{1≤i≤N} W_{S}^{u}(π_{i})

b4. W_{S}^{s}(π_{i}) and W_{S}^{u}(π_{j}) are transversal for i, j ∈ {1, …, N}.

It is known that S(t) has Property b generically with respect to the nonlinearity f. Let us explain the meaning of this statement.

Fix an integer q ≥ 0. We introduce the C^{q} strong Whitney topology on the set of functions f : R → R of class C^{q} as follows. For any functions f, g ∈ C^{q}(R) and for any compact set K ⊂ R define the number

ρ_{K}^{q}(f, g) = ∑_{r=0}^{q} sup_{v∈K} |f^{(r)}(v) − g^{(r)}(v)|.

The base of neighborhoods of a function f in the C^{q} strong Whitney topology consists of the sets

{g : ρ_{K_{n}}^{q}(f, g) < ϵ_{n}},

where {K_{n}} is a countable family of compact subsets of R such that any point of R belongs to a finite number of the sets K_{n}, the equality ⋃_{n} K_{n} = R holds, and {ϵ_{n}} is a sequence of positive numbers.

A subset Y of a topological space X is called residual if Y contains a countable intersection of open and dense subsets of X. If P is a property of elements of X, then we say that this property is generic if the set {x ∈ X : x satisfies P } is residual. It is known [13] that every residual subset of the space of functions f : R → R of class C^{q} with the C^{q} strong Whitney topology is dense in this space.

Brunovsky and Chow [6] showed that there exists a residual subset G of the set of functions of class C^{2} on R with the C^{2} strong Whitney topology having the following two properties.

• If f ∈ G, then any solution p(x) of (8) is hyperbolic;

• if f ∈ G, then the set of μ > 0 for which the problem

p_{xx} + μf(p) = 0, p(0) = p(1) = 0, (9)

has nonhyperbolic solutions is countable. For a special class of functions f, it is possible to give an explicit description of

the set {μ} that corresponds to nonhyperbolic solutions of problem (9). Chaffee and Infante [7] considered (1) with a nonlinearity μf ∈ C^{2} and f satisfying the conditions

• f(0) = 0, f′(0) = 1;

• uf′′(u) < 0 for u ≠ 0;

• \overline{lim}_{u→±∞}f(u)/u ≤ 0.

It is shown in [12] that problem (9) has only hyperbolic solutions if μ ≠ n^{2}π^{2}, n = 1, 2, . . ..

Thus, for a C^{2}-generic nonlinearity all fixed points of S(t) are hyperbolic. It follows from results of Henry [12] and Angenent [3] that for hyperbolic fixed points their stable and unstable manifolds are always transversal. Hence, for a generic nonlinearity f in (1) satisfying condition (6), the global attractor A has properties b1 and b4. It is easy to establish also properties b2 and b3.

Let σ(u) = S(1)u for u ∈ H. Our goal is to study conditions under which the semi-dynamical system σ has a variant of the Lipschitz shadowing property on a neighborhood of the global attractor of S(t). Fix d > 0. As usual, we say that a sequence {u_{j} ∈ H : j ≥ 0} is a d-pseudotrajectory of σ if

|σ(u_{j}) − u_{j+1}| ≤ d for j ≥ 0. (10)

We say that the system σ has a Lipschitz shadowing property on a set W if there exist positive constants d_{0}, L_{0} such that for any d-pseudotrajectory {u_{j}} ⊂ W with d ≤ d_{0} there is a point u such that

|σ^{j}(u) − u_{j}| ≤ L_{0}d for j ≥ 0. (11)

Our proof of the shadowing property is based on the existence of a smooth inertial manifold constructed as follows. There exists an orthogonal projection P with finite-dimensional range P H, and a C^{1}-mapping Φ : P H → QH (where Q = I − P) such that M, the graph of Φ, has the following two properties.

• S(t)M ⊂ M, t ≥ 0;

• M is exponentially attractive, i.e., for any bounded set B ⊂ H there exist positive constants C, a (depending on B) such that

(12) dist(S(t)u, M) ≤ Ce^{−at}dist(u, M), for u ∈ B, t ∈ [0, ∞).

Obviously, it follows from our definitions that A ⊂ M. It is known [9], [8] that if the nonlinearity F in (4) is modified so that it vanishes outside a ball in H, then the corresponding modified semigroup has an inertial manifold.

Fix an absorbing set H_{0} for A. It is known that there is a bounded set H′ such that

S(t)H_{0} ⊂ H′ for t ≥ 0;

cf. (5). We denote by H^{∗} the 1-neighborhood of H′ and modify F so that the new nonlinearity coincides with F in H^{∗} and vanishes on a neighborhood of infinity. We keep the notation F and S(t) for the modified system. It follows that the new system has an inertial manifold M and its trajectories beginning at points of H_{0} coincide with the corresponding trajectories of the original system. Let inequality (12) be satisfied for u ∈ H^{∗}.

Theorem 1. Assume that S(t) has Properties a and b. Then there exist constants d_{0}, L_{0} > 0 and a neighborhood W of A in H such that if {u_{j} : j ≥ 0} ⊂ W is a d-pseudotrajectory of σ with d ≤ d_{0} (see (10)), and

dist(u_{0}, M) ≤ 2d, (13)

then there is a point u ∈ M such that (11) holds.

To prove this theorem, we first reduce the shadowing problem to an analogous problem on the finite-dimensional manifold M, and then establish the desired shadowing property on M.

The assumption (13), which is not needed in the finite-dimensional case (see [19] and Theorem 2 below), occurs in the reduction; see (19). We do not know if it is really necessary in the infinite-dimensional case.

3. Reduction to the Inertial Manifold

Take an integer T > 0 such that, with C, a as in (12),

ν := Ce^{−aT} < \frac{1}{2}. (14)

Below we impose another restriction on T; this restriction is “absolute”, i.e., it depends only on the attractor A; see (33).

For u_{0} ∈ H let p_{0} = P u_{0}. Denote by p(t, p_{0}) the solution of the finite-dimensional system in P H,

ṗ + Ap = P F (p + Φ(p)), t > 0; p(0, p_{0}) = p_{0}. (15)

Then (see [9]) for u_{0} ∈ M we have

u(t) = S(t)u_{0} = p(t, p_{0}) + Φ(p(t, p_{0})). (16)

We introduce the following notation. For p_{0} = P u_{0}, u_{0} ∈ H we set

σ_{1}^{∗}(p_{0}) = p(1, p_{0}), σ_{T}^{∗} (p_{0}) = p(T, p_{0})

(the integer T was fixed in (14)), and for m_{0} ∈ M we set p_{0} = P m_{0} and

ϕ_{1}(m_{0}) = σ_{1}^{∗}(p_{0}) + Φ(σ_{1}^{∗}(p_{0})), ϕ(m_{0}) = σ_{T}^{∗} (p_{0}) + Φ(σ_{T}^{∗} (p_{0})), (17)

It follows from the inclusions f, Φ ∈ C^{1} that σ_{1}^{∗}, σ_{T}^{∗} (and ϕ_{1}, ϕ) are C^{1}-mappings of P H (of M, respectively) to itself. In addition, by (16) for u_{0} ∈ M we have

ϕ_{1}(u_{0}) = S(1)u_{0} = σ(u_{0}),

i.e., ϕ_{1} is the restriction of σ to M.

Obviously, A is the global attractor of σ in H and of ϕ_{1} in M (the definitions are parallel to the one for S).

Set C_{1} = C(1, H_{0}), C_{T} = C(T, H_{0}); see (7). Since the system (15) is linear in a neighborhood of infinity, all its solutions are defined for t ∈ R. It follows that ϕ, ϕ^{−1} are diffeomorphisms of M of class C^{1}.

Fix a neighborhood M_{0} ⊂ H_{0} of A in M. Since A is Lyapunov stable, there exists a neighborhood M_{1} ⊂ M_{0} of A such that S(t)u ∈ M_{0} for u ∈ M_{1} and t ≥ 0. It follows that for p_{0} ∈ Π_{1} = P M_{1} we have p(t, p_{0}) ∈ Π_{0} = P M_{0} for t ≥ 0. Let K_{1} be a Lipschitz constant of σ_{T}^{∗} = p(T, ·) on Π_{1}.

Due to the same reason there exists d^{0} > 0 such that the inequality dist(u, A) ≤ d^{0} implies the inclusions σ^{n}(u) ∈ H_{0} for n ≥ 0. Let W_{1} be the d^{0}-neighborhood of A. Assume, in addition, that the d^{0}-neighborhood of W_{1} is a subset of H_{0} and that P projects this neighborhood of W_{1} into Π_{1}.

The following Lemmas 1 and 2 show that our problem of Lipschitz shadowing for d-pseudo-trajectories of σ is reduced to the same problem for Cd-pseudotrajectories of σ^{T}.

Take a d-pseudotrajectory {u_{j}} ⊂ W_{1} of σ with d ≤ d^{0}. Let z_{j} = u_{jT}.

Lemma 1. Let C_{2} = 1 + C_{1} + ⋯ + C_{1}^{T −1}, C_{3} = C_{2}/(1 − ν). Then (i) the sequence {z_{j}} is a C_{2}d-pseudotrajectory of σ^{T}; (ii) if dist(z_{0}, M) ≤ 2d, then dist(z_{j}, M) ≤ 2C_{3}d for j ≥ 0.

Proof. First note that the choice of d^{0} and W_{1} implies that σ^{n}(u_{j}) ∈ H_{0} for n ≥ 0. Let us prove statement (i). By definition,

|σ(z_{j}) − u_{jT +1}| = |σ(u_{jT}) − u_{jT +1}| ≤ d. (18)

Now we estimate

|σ^{2}(z_{j}) − u_{jT +2}| ≤ |σ(u_{jT +1}) − u_{jT +2}| + |σ(σ(u_{jT})) − σ(u_{jT +1})|
≤ d + C_{1}d,

since both u_{jT +1} and σ(u_{jT}) are in H_{0}, and (18) holds. Similarly,

|σ^{3}(z_{j}) − u_{jT +3}| ≤ |σ(u_{jT +2}) − u_{jT +3}| + |σ(σ^{2}(u_{jT})) − σ(u_{jT +2})|
≤ d + C_{1}(d + C_{1}d) = d + C_{1}d + C_{1}d^{2},

and so on. Thus, we arrive at the inequality

|σ^{T} (z_{j}) − z_{j+1}| ≤ d(1 + C_{1} + ⋯ + C_{1}^{T −1}) = C_{2}d.

This shows that {z_{j}} is a C_{2}d-trajectory of σ^{T}.

To prove statement (ii), let b_{j} = dist(z_{j}, M). By assumption, we have b_{0} ≤ 2d. Now we estimate

b_{j+1} ≤ dist(σ^{T} (z_{j}), z_{j+1}) + dist(σ^{T} (z_{j}), M)
≤ C_{2}d + νdist(z_{j}, M) = C_{2}d + νb_{j};

see (12) and (14). It follows that

b_{0} ≤ 2d ≤ 2C_{2}d, b_{1} ≤ C_{2}d + νb_{0} ≤ 2C_{2}d(1 + ν), …,
b_{n} ≤ 2C_{2}d(1 + ν + ⋯ + ν^{n}) ≤ \frac{2C_{2}d}{1 − ν} .

This completes the proof.

Lemma 2. Assume that for some u ∈ H we have

σ^{n}(u) ∈ H_{0}, n ≥ 0, and |σ^{jT} (u) − z_{j}| ≤ L′d, j ≥ 0.

Then

|σ^{j}(u) − u_{j}| ≤ C_{5}d, j ≥ 0,

where C_{5} = 1 + C_{4} + ⋯ + C_{4}^{T}, C_{4} = C_{1}L′.

Proof. Take n ∈ (jT, (j + 1)T]. One easily obtains the inequalities

|σ^{jT +1}(u) − u_{jT +1}| ≤ |σ^{jT +1}(u) − σ(u_{jT})| + |σ(u_{jT}) − u_{jT +1}|
≤ C_{1}|σ^{jT} (u) − u_{jT} | + d ≤ C_{4}d + d = d(1 + C_{4}).

Similarly,

|σ^{jT +2}(u) − u_{jT +2}| ≤ |σ(σ^{jT +1}(u)) − σ(u_{jT +1})| + |σ(u_{jT +1}) − u_{jT +2}|
≤ C_{1}(C_{4}d + d) + d ≤ d(1 + C_{4} + C_{4}^{2}).

Finally, we arrive at the desired estimate

|σ^{n}(u) − u_{n}| ≤ d(1 + C_{4} + ⋯ + C_{4}^{T}).

The following lemma will be used in Section 5 to show that assumption (13) can be satisfied.

Lemma 3. Let the ball B = {u ∈ H : |u| ≤ R} be a subset of H_{0}. For any d > 0 there exists a number N = N(R, d) such that if {u_{j}} ⊂ H_{0} is a d-pseudotrajectory of σ^{T} with u_{0} ∈ B, then dist(u_{n}, M) ≤ 2d for some n ∈ [0, N].

Proof. Let b_{j} = dist(u_{j}, M) and set ν_{1} = ν + \frac{1}{2}. It follows from (14) that ν_{1} < 1. Select N > 0 such that ν_{1}^{N}R < 2d.

To obtain a contradiction, assume that b_{j} > 2d for 0 ≤ j ≤ N. Then for 0 ≤ j ≤ N − 1 we have, by (12) and (14),

b_{j+1} ≤ dist(σ^{T} (u_{j}), M) + |σ^{T} (u_{j}) − u_{j+1}| ≤ νb_{j} + d.

By our assumption, b_{j} > 2d, we have d < b_{j}/2, and hence

b_{j+1} ≤ (ν + \frac{1}{2})b_{j} = ν_{1}b_{j} for 0 ≤ j ≤ N − 1.

It follows that

b_{N} ≤ ν_{1}^{N}b_{0} ≤ ν_{1}^{N}R < 2d.

The obtained contradiction proves the lemma.

We continue to discuss the sequence {z_{j}} introduced before Lemma 1. Take d^{1} = d^{0}/(2C_{3}). We assume in the sequel that d ≤ d^{1}.

Let z′_{j} = P z_{j} and set v_{j} = z′_{j} + Φ(z′_{j}) ∈ M. We shall prove that {v_{j}} is a C′d-pseudotrajectory of ϕ, defined in (17), with C′ independent of d.

Since dist(z_{j}, M) ≤ 2C_{3}d (by Lemma 1), we can find w_{j} ∈ M such that

|w_{j} − z_{j}| ≤ 2C_{3}d. (19)

We obtain from the choice of d^{1} that w_{j} ∈ H_{0}. Now it follows from (7) and (19) that

|σ^{T} (w_{j}) − σ^{T} (z_{j})| ≤ C_{6}d, (20)

where C_{6} = 2C_{T} C_{3}, C_{T} = C(T, H_{0}). From Lemma 1 and from (20) we conclude that

|σ^{T} (w_{j}) − z_{j+1}| ≤ |σ^{T} (z_{j}) − z_{j+1}| + |σ^{T} (w_{j}) − σ^{T} (z_{j})| ≤ C_{7}d, (21)

where C_{7} = C_{2} + C_{6}. Let w′_{j} = P w_{j}. By our choice, z′_{j}, w′_{j} ∈ Π_{1}. It follows from the definition of ϕ in (17) that

P σ^{T} (w_{j}) = P ϕ(w_{j}) = σ_{T}^{∗} (P w_{j}) = σ_{T}^{∗} (w′_{j}). (22)

We want to estimate the value

|σ_{T}^{∗} (z′_{j}) − z′_{j+1}| ≤ |σ_{T}^{∗} (z′_{j}) − σ_{T}^{∗} (w′_{j})| + |σ_{T}^{∗} (w′_{j}) − P z_{j+1}|. (23)

The first term in (23) does not exceed K_{1}C_{3}d, because K_{1} is a Lipschitz constant of σ_{T}^{∗} on Π_{1} and

|z′_{j} − w′_{j}| = |P z_{j} − P w_{j}| ≤ C_{3}d;

see (19)). By (22) we have

|σ_{T}^{∗} (w′_{j}) − P z_{j+1}| = |P σ^{T} (w_{j}) − P z_{j+1}|.

Hence, the second term in (23) does not exceed

|σ^{T} (w_{j}) − z_{j+1}| ≤ C_{7}d;

see (21). Thus, we obtain the estimate

|σ_{T}^{∗} (z′_{j}) − z′_{j+1}| ≤ C_{8}d, (24)

where C_{8} = K_{1}C_{3} + C_{7}.

Let K_{2} be a Lipschitz constant of Φ on Π_{1}. We obtain from (24) the estimate

|v_{j+1} − ϕ(v_{j})| ≤ |z′_{j+1} − σ_{T}^{∗} (z′_{j})| + |Φ(z′_{j+1}) − Φ(σ_{T}^{∗} (z′_{j}))| ≤ C_{9}d, (25)

where C_{9} = (1 + K_{2})C_{8}.

Denote by r the metric on M generated by the distance in H. Obviously, there is a constant K_{3} > 0 such that

K_{3}^{−1}r(v, v′) ≤ |v − v′| ≤ r(v, v′) for v, v′ ∈ M_{1}.

It follows from (25) that r(v_{j+1}, ϕ(v_{j})) ≤ K_{3}C_{9}d. Thus, {v_{j}} is a K_{3}C_{9}dpseudotrajectory of ϕ on M. Let us show that Theorem 1 follows from the following statement.

Theorem 2. There exists a neighborhood M of A in M and numbers d′, L > 0 such that if {v_{j}} ⊂ M is a d-pseudotrajectory of ϕ with d ≤ d′, then there is a point v ∈ M with the property

r(ϕ^{j}(v), v_{j}) ≤ Ld, j ≥ 0.

Indeed, let {u_{j}} be as in Theorem 1, define {z_{j}}, ϕ, and {v_{j}} as above. By using Lemma 1 we showed that {v_{j}} is a C′d-pseudotrajectory of ϕ on M. Theorem 2 then gives v ∈ M such that

r(ϕ^{j}(v), v_{j}) ≤ L′d, j ≥ 0 (L′ = LC′d),

and hence

|ϕ^{j}(v) − v_{j}| ≤ L′d, j ≥ 0. (26)

Let us estimate

|z_{j} − v_{j}| ≤ |z_{j} − w_{j}| + |w_{j} − v_{j}|.

By (19), the first term (and also |z′_{j} − w′_{j}|) does not exceed 2C_{3}d. Since w_{j} = w′_{j} + Φ(w′_{j}), v_{j} = z′_{j} + Φ(z′_{j}), we estimate the second term by 2C_{3}d(1 + K_{2}). Now we obtain from (26) the inequality

|z_{j} − σ^{jT} (v)| = |z_{j} − ϕ^{j}(v)| ≤ (L′ + 2C_{3}(2 + K_{2}))d = L′′d.

Lemma 2 now implies that |σ^{j}(v)−u_{j}| ≤ C_{5}L′′d = L′′′d. This proves our reduction statement.

4. Shadowing on the Inertial Manifold

In this section we prove Theorem 2.

Properties b1–b4 of the attractor A of σ imply the following properties of the mapping ϕ_{1} (defined in (17)) on M. b1’. A contains fixed points π_{1}, …, π_{N} of ϕ_{1}, and they are hyperbolic (we denote

by W^{s}(π_{i}), W^{u}(π_{i}) their stable and unstable manifolds); b2’.

M = ⋃_{1≤i≤N} W^{s}(π_{i}); (27)

b3’.

A = ⋃_{1≤i≤N} W^{u}(π_{i}); (28)

b4’. W^{s}(π_{i}) and W^{u}(π_{j}) are transversal for i, j ∈ {1, …, N}.

Let us prove only b4’. Take a point

x ∈ W^{s}(π_{i}) ∩ W^{u}(π_{j}).

As before, we denote by T_{p}N the tangent space of N at p. Let

\begin{aligned}T_{1}^{s} & = T_{x}W^{s}(π_{i}), & T_{2}^{s} = T_{x}W_{S}^{s}(π_{i}), \\ T_{1}^{u} & = T_{x}W^{u}(π_{j}), & T_{2}^{u} = T_{x}W_{S}^{u}(π_{j}). \end{aligned}

By b4, W_{S}^{s}(π_{i}) and W_{S}^{u}(π_{j}) are transversal at x, i.e.,

T_{2}^{s} ⊕ T_{2}^{u} = H. (29)

We claim that

T_{1}^{z} = T_{x}M ∩ T_{2}^{z}, z = s, u. (30)

Note that the norm in T_{p}M, p ∈ M, generated by the metric r on M coincides with the norm in T_{p}M induced from H.

The equality

T_{1}^{s} = {w ∈ T_{p}M : |Dϕ_{1}^{k}(x)w| → 0, k → ∞}

is proved in [17, Chap. 13]. Similarly one proves that

T_{2}^{s} = {w ∈ H : |Dσ^{k}(x)w| → 0, k → ∞}.

Obviously these equalities imply (30) for z = s. For z = u the proof is similar. Now it follows from (29) that

T_{1}^{s} ⊕ T_{1}^{u} = T_{x}M,

i.e., W^{s}(π_{i}) and W^{u}(π_{j}) are transversal at x.

Since any fixed point π_{i} of ϕ_{1} is hyperbolic, there exist the “stable” and “unstable” subspaces E_{i}^{s} and E_{i}^{u} of T_{π_{i}}M with the standard properties,

• E_{i}^{z}, z = s, u, are Dϕ_{1}-invariant;

• E_{i}^{s} ⊕ E_{i}^{u} = T_{π_{i}}M;

• there are K_{i} > 0, λ_{i} ∈ (0, 1) such that

(31)  |Dϕ_{1}^{m}(π_{i})v| ≤ K_{i}λ_{i}^{m}|v|, v ∈ E_{i}^{s}, m ≥ 0,
(32)  |Dϕ_{1}^{−m}(π_{i})v| ≤ K_{i}λ_{i}^{m}|v|, v ∈ E_{i}^{u}, m ≥ 0.

Take λ_{0} ∈ (0, 1) and an integer T_{0} such that

K_{i}λ_{i}^{T_{0}} < λ_{0}, 1 ≤ i ≤ N. (33)

We fix T satisfying (14) so that T ≥ T_{0} (note that T_{0} is an “absolute” constant). Since ϕ = ϕ_{1}^{T} (see (17)), it follows from (31) and (32) that

∥Dϕ|_{E_{i}^{s}}∥ ≤ λ_{0}, ∥Dϕ^{−1}|_{E_{i}^{u}}∥ ≤ λ_{0}. (34)

For a set X ⊂ M we denote

O^{+}(X) = ⋃_{k≥0} ϕ^{k}(X), O^{−}(X) = ⋃_{k≤0} ϕ^{k}(X), O(X) = O^{+}(X) ∪ O^{−}(X).

The proof of the following statement repeats the proof of parts I, III, and IV of [20, Theorem 10.1]. We denote by TM the tangent bundle of M.

Lemma 4. There exists λ_{1} ∈ (λ_{0}, 1), neighborhoods Z_{i} of π_{i} in M, and continuous subbundles S_{i}, U_{i} of TM on \overline{Z}_{i} ∪ O(Z_{i}) such that

(i) S_{i}(x) ⊕ U_{i}(x) = T_{x}M for x ∈ Z_{i}; (ii) S_{i}, U_{i} are Dϕ-invariant on O(Z_{i}) (consequently, the equalities in (i) hold for

x ∈ O(Z_{i})); (iii)

∥Dϕ|_{S_{i}(x)}∥ ≤ λ_{1}, ∥Dϕ^{−1}|_{U_{i}(x)}∥ ≤ λ_{1} for x ∈ Z_{i};

(iv)

S_{i}(x) ⊂ S_{j}(x), U_{j}(x) ⊂ U_{i}(x) for x ∈ O^{+}(Z_{i}) ∩ O^{−}(Z_{j}).

It is shown in [18] that there exists a neighborhood M_{2} ⊂ M_{1} of A such that ϕ(\overline{M}_{2}) ⊂ M_{2}. We define

Z = ⋃_{1≤i≤N} Z_{i}.

Let τ be a Birkhoff constant for ϕ on \overline{M}_{2}, i.e., a positive number with the property

card\{k ∈ Z : ϕ^{k}(x) ∈ \overline{M}_{2} \ Z\} ≤ τ

for x ∈ \overline{M}_{2} (here card is the cardinality). Let M_{3} = ϕ(M_{2}).

Let

N_{0} = max_{x∈\overline{M}_{2}}(∥Dϕ(x)∥, ∥Dϕ^{−1}(x)∥).

Take p ∈ \overline{M}_{2}, there exists m_{0} ∈ [0, τ] such that q = ϕ^{m_{0}}(p) ∈ Z_{i} and ϕ^{m}(p) ∉ Z_{l} with i ≠ l for 0 ≤ m < m_{0}. Define linear subspaces S(p), U(p) of T_{p}M by

S(p) = Dϕ^{−m_{0}}(q)S_{i}(q), U(p) = Dϕ^{−m_{0}}(q)U_{i}(q).

Now we define a mapping e_{p} : T_{p}M → M for p ∈ M by

e_{p}(v) = P (p + v) + Φ(P (p + v)), v ∈ T_{p}M.

Since Φ is of class C^{1}, e_{p} is also of class C^{1}.

Straightforward calculation shows that

De_{p}(0) = I. (35)

The following statement is proved similarly to [19, Lemma 2.1].

Lemma 5. For p ∈ \overline{M}_{2} the spaces S(p), U(p) have the properties (i) S(p) ⊕ U(p) = T_{p}M; (ii) There exists N_{1} > 0 such that for Q^{s}(p), Q^{u}(p), the projectors onto S(p), U(p)

parallel to U(p), S(p), respectively, the inequalities

∥Q^{s}(p)∥ ≤ N_{1}, ∥Q^{u}(p)∥ ≤ N_{1}

hold; (iii)

Dϕ(p)S(p) ⊂ S(ϕ(p)), Dϕ^{−1}(p)U(p) ⊂ U(ϕ^{−1}(p))

(the second inclusion holds if, in addition, p ∈ \overline{M}_{3}); (iv) for each μ, a_{1} > 0 there exists a_{2} > 0 such that if p, z ∈ ϕ^{μ}(\overline{M}_{2}) and

r(z, ϕ^{μ}(p)) < a_{2}, then there is a linear isomorphism Π(p, z) : T_{z}M → T_{z}M with the properties

∥Π(p, z) − I∥ < a_{1}, Π(p, z)[De_{z}^{−1}(q)Dϕ^{μ}(p)S(p)] ⊂ S(z),

and a linear isomorphism Θ(p, z) : T_{p}M → T_{p}M with the properties

∥Θ(p, z) − I∥ < a_{1}, Θ(p, z)[De_{p}^{−1}(t)Dϕ^{−μ}(z)U(z)] ⊂ U(p),

where q = ϕ^{μ}(p), t = ϕ^{−μ}(z).

It is easy to see that for v ∈ S_{i}(x) we have

|Dϕ^{m}(x)v| ≤ Kλ_{1}^{m}|v|, m ≥ 0,

and for v ∈ U_{i}(x) we have

|Dϕ^{m}(x)v| ≤ Kλ_{1}^{−m}|v|,

if m ≤ 0 and ϕ^{k}(x) ∈ \overline{M}_{2} for k ∈ [m, 0], where K = (N_{0}/λ_{1})^{τ} (and τ is the Birkhoff constant chosen above). Take μ ≥ τ such that Kλ_{1}^{μ} < λ_{1}. Let ξ = ϕ^{μ}.

The same reasons as above (see Lemmas 1 and 2) show that ϕ has a Lipschitz shadowing property in a neighborhood of A if and only if ξ has. Thus, we work below with ξ.

Obviously, Lemma 5 remains true for ξ instead of ϕ, but in statement (iii) M_{3} is to be replaced by M_{4}, where

M_{4} = ξ(M_{2}).

In addition, by the choice of μ, the following inequalities hold:

|Dξ(p)v| ≤ λ_{1}|v|, v ∈ S(p), p ∈ M_{2}, (36)

|Dξ^{−1}(p)v| ≤ λ_{1}|v|, v ∈ U(p), p ∈ M_{4}. (37)

For a > 0 we denote

E_{a}(p) = {v ∈ T_{p}M : |v| < a}, D_{a}(p) = e_{p}(E_{a}(p)).

Obviously, there exists a neighborhood M_{5} ⊂ M_{4} of A and a number c > 0 such that for p ∈ \overline{M}_{5} we have

D_{c}(p), ξ(D_{c}(p)), ξ^{−1}(D_{c}(p)) ⊂ M_{4},

and e_{p} is a diffeomorphism of E_{c}(p) onto D_{c}(p) with uniform estimates of ∥De_{p}∥ and ∥De_{p}^{−1}∥.

Let M = M_{5}. Take a d-pseudotrajectory {v_{k} : k ≥ 0} ⊂ M of ξ.

Below we denote by d′ positive constants that depend on properties of ξ on M and do not depend on {v_{k}}. At each step of the proof, we consider d-pseudotrajectories such that d does not exceed the minimal d′ previously chosen. Since we choose d′ finitely many times, no generality is lost.

In the proof below we apply a “local” shadowing result for a sequence of mappings of Banach spaces ([19, Theorem 1.1]). Let us formulate it. Consider a sequence of Banach spaces H_{k}, k ≥ 0. We denote by | · | norms in H_{k} and by ∥ · ∥ the corresponding operator norms for linear operators. Consider a sequence of mappings

ϕ_{k} : H_{k} → H_{k+1}

of the form

ϕ_{k}(u) = A_{k}u + w_{k+1}(u),

where A_{k} are linear mappings.

Lemma 6. Assume that

(i) there exist numbers λ ∈ (0, 1), N > 0, and projectors P_{k}, Q_{k} in H_{k} (we denote below S_{k} = P_{k}H_{k}, U_{k} = Q_{k}H_{k}) such that

∥P_{k}∥, ∥Q_{k}∥ ≤ N, P_{k} + Q_{k} = I,
∥A_{k}|_{S_{k}}∥ ≤ λ, A_{k}S_{k} ⊂ S_{k+1};

NUMERICAL SHADOWING NEAR THE GLOBAL ATTRACTOR 13

(ii) there exist linear mappings B_{k} : U_{k+1} → H_{k} such that

B_{k}U_{k+1} ⊂ U_{k}, ∥B_{k}∥ ≤ λ, A_{k}B_{k}|_{U_{k+1}} = I;

(iii) there exist numbers κ, Δ > 0 such that

|w_{k+1}(u) − w_{k+1}(u′)| ≤ κ|u − u′| for |u|, |u′| ≤ Δ,

and κN′ < 1, (38)

where N′ = N \frac{1+λ}{1−λ}. Then there exist constants d^{∗}, L^{∗} > 0 (depending on λ, N, Δ) with the following

property: if

|ϕ_{k}(0)| ≤ d ≤ d^{∗} for k ≥ 0,

then one can find points u_{k} ∈ H_{k} such that ϕ_{k}(u_{k}) = u_{k+1} and

|u_{k}| ≤ L^{∗}d.

Now we fix k ≥ 0 and denote p = v_{k}, z = v_{k+1}, H_{k} = T_{v_{k}}M. Take d′ < c/2 and such that the inequality r(z, ξ(p)) ≤ d′ (39) implies the inclusion ξ^{−1}(z) ∈ D_{c}(p). (40) Find 0 < b < c such that

ξ(D_{b}(x)) ⊂ D_{d′}(ξ(x)), x ∈ M

(obviously, b depends only on ξ). Then it follows from (39) that

ξ(D_{b}(p)) ⊂ D_{c}(z).

Thus, the mapping ψ_{k} : E_{b}(p) → H_{k+1} given by

ψ_{k}(w) = e_{z}^{−1} ∘ ξ ∘ e_{p}(w)

is properly defined. Let us introduce the following notation:

q = ξ(p), q′ = e_{z}^{−1}(q), t = ξ^{−1}(z), t′ = e_{p}^{−1}(t),

D = Dψ_{k}(0), D′ = Dξ(t)De_{p}(t′), G = De_{p}^{−1}(t)Dξ^{−1}(z).

Note that t′, D′, G are well-defined since (40) holds.

Fix N_{2} > 0 such that

∥Dξ(x)∥, ∥Dξ^{−1}(x)∥ ≤ N_{2}, x ∈ M.

Let N = max(N_{1}, N_{2}) (N_{1} is given by Lemma 5).

Find ν_{0} ∈ (0, 1) such that

λ = (1 + ν_{0})^{2}λ_{1} < 1

and let

N′ = N \frac{1 + λ}{1 − λ}.

Take κ > 0 such that (38) holds and find ν < ν_{0} with the property N(3N + 1)ν < \frac{1}{2}κ. (41)

14 STIG LARSSON AND SERGEI YU. PILYUGIN

It follows from (35) that D = De_{z}^{−1}(q)Dξ(p). (42)

Now we find d′ such that inequality (39) implies the inequalities

(43)  |De_{p}(t′)w| ≤ (1 + ν)|w|, w ∈ T_{p}M,
(44)  |De_{z}^{−1}(q)w| ≤ (1 + ν)|w|, w ∈ T_{q}M,
(45)  |De_{p}^{−1}(t)w| ≤ (1 + ν)|w|, w ∈ T_{t}M

(we apply (35) and the uniform continuity of De_{x}, De_{x}^{−1}), ∥D − D′∥ < ν (46) (compare (42)), the definition of D′, and take into account the previous argument), ∥Π(p, z) − I∥, ∥Θ(p, z) − I∥, ∥Θ^{−1}(p, z) − I∥ < ν; (47) see Lemma 5.

Define A_{k} : H_{k} → H_{k+1} by

A_{k} = Π(p, z)DQ^{s}(p) + D′Θ^{−1}(p, z)Q^{u}(p).

For w ∈ S(p) we have w^{s} = A_{k}w = Π(p, z)Dw ∈ S(z) (see Lemma 5), hence A_{k}S(p) ⊂ S(z). (48) Since |Dξ(p)w| ≤ λ_{1}|w| by (36), it follows from (44) and (47) that

|w^{s}| ≤ (1 + ν)^{2}λ_{1}|w| = λ|w|.

Hence ∥A_{k}|_{S(p)}∥ ≤ λ. (49)

Now we consider a mapping B_{k} : U(z) → H_{k} defined as follows: B_{k}w = Θ(p, z)Gw. We easily show that w^{u} = B_{k}w ∈ U(p), hence B_{k}U(z) ⊂ U(p), (50) and that |w^{u}| ≤ λ|w|. Hence ∥B_{k}|_{U(z)}∥ ≤ λ. (51) It follows from

A_{k}w^{u} = D′Θ^{−1}(p, z)Θ(p, z)Gw = D′Gw = w

that A_{k}B_{k}|_{U(z)} = I. (52) Represent

ψ_{k}(w) = Dw + ρ(w).

Obviously there exists d′ such that |ρ(w) − ρ(w′)| ≤ \frac{1}{2}κ|w − w′| (53) for |w|, |w′| ≤ d′.

Now we represent ψ_{k}(w) = A_{k}w + χ(w), (54) where χ(w) = (D − A_{k})w + ρ(w), χ(0) = ψ_{k}(0).

Let us estimate ∥D − A_{k}∥. We have

∥D − A_{k}∥ = ∥D(Q^{s}(p) + Q^{u}(p)) − A_{k}∥
≤ ∥DQ^{s}(p) − Π(p, z)DQ^{s}(p)∥ + ∥DQ^{u}(p) − D′Q^{u}(p)∥
+ ∥D′Q^{u}(p) − D′Θ^{−1}(p, z)Q^{u}(p)∥.

Since ∥Q^{s}(p)∥, ∥D∥ ≤ N, and ∥Π(p, z) − I∥ < ν (see (47)), we obtain that the first term on the right does not exceed N^{2}ν. The same reasons (and inequality (46)) show that the second term is estimated by Nν. From the definition of D′ and from the inequality ν < 1 we obtain that ∥D′∥ < 2N. Thus, the third term is estimated by 2N^{2}ν.

This gives the inequality

∥D − A_{k}∥ ≤ N(3N + 1)ν ≤ \frac{1}{2}κ;

see (41). Combined with (53), the last inequality shows that

|χ(w) − χ(w′)| ≤ κ|w − w′| for |w|, |w′| ≤ d′. (55)

Since the derivatives De_{x}(y), De_{x}^{−1}(y′) are uniformly bounded for

x ∈ M, y ∈ E_{c}(x), y′ ∈ D_{c}(x), (56)

there exists N^{∗} > 0 such that for x, y, y′ that satisfy (56) we have

r(x, e_{x}(y)) ≤ N^{∗}|x − y|, |x − e_{x}^{−1}(y′)| ≤ N^{∗}r(x, y′).

Since ψ_{k}(0) = e_{z}^{−1}(ξ(p)), it follows from r(ξ(p), z) < d that

|ψ_{k}(0)| ≤ N^{∗}r(z, ξ(p)) ≤ N^{∗}d. (57)

Now we obtain from Lemma 5 and from (48)–(52), (55), and (57) that ψ_{k} satisfy all the conditions of Lemma 6 (with the obvious change of notation).

Hence, there exist d′, L′ > 0 (depending only on ξ and M) with the following property: if {v_{k}} is a d-pseudotrajectory of ξ with d ≤ d′, then there exists a sequence w_{k} ∈ H_{k} such that ψ_{k}(w_{k}) = w_{k+1} and |w_{k}| ≤ L′d.

Set w′_{k} = e_{v_{k}}(w_{k}). Then it follows from the definition of ψ_{k} that w′_{k+1} = ξ(w′_{k}), hence the sequence w′_{k} is a ξ-trajectory of v = w′_{0}, i.e., w′_{k} = ξ^{k}(v).

Now we obtain from the definition of N^{∗} that there exists d_{0} such that, if {v_{k}} is a d-pseudotrajectory of ξ with d ≤ d_{0}, then the inequalities

r(v_{k}, ξ^{k}(v)) = r(v_{k}, w′_{k}) ≤ N^{∗}|w_{k}| ≤ Ld,

where L = N^{∗}L′, hold for k ≥ 0. This completes the proof of Theorem 2.

5. The Discretized Problem

In this section we show how to apply the shadowing result when the pseudotrajectory is a numerical approximation defined by means of a standard piecewise linear finite element method and the backward Euler time stepping.

The finite element method is based on the weak formulation of the initial boundary value problem (1),

\begin{matrix}u(t) ∈ H, u(0) = u_{0}, \\ (u_{t}, v) + (u_{x}, v_{x}) = (f(u), v), ∀v ∈ H, t > 0, \end{matrix}

obtained by integration by parts, taking the boundary condition into account; the inner product was defined in (2). Let {H_{h}}_{h>0} denote a family of finite dimensional subspaces of H, where each H_{h} consists of continuous piecewise polynomials of degree ≤ 1 with respect to a mesh 0 = x_{0} < x_{1} < ⋯ < x_{M} = 1, and where h = max(x_{i} − x_{i−1}). The backward Euler method with constant step k and times t_{j} = jk, j = 0, 1, …, is

\begin{matrix}U_{j} ∈ H_{h}; U_{0} = v_{h}, \\ k^{−1}(U_{j} − U_{j−1}, χ) + (U_{j,x}, χ_{x}) = (f(U_{j}), χ), ∀χ ∈ H_{h}, t_{j} > 0. \end{matrix} (58)

Introducing the operator A_{h} : H_{h} → H_{h} and the orthogonal projector P_{h} : L_{2} → H_{h} defined by

(A_{h}ψ, χ) = (ψ_{x}, χ_{x}), (P_{h}ϕ, χ) = (ϕ, χ), ∀ψ, χ ∈ H_{h}, ϕ ∈ L_{2},

we may write (58) as

\begin{matrix}U_{j} ∈ H_{h}; U_{0} = v_{h}; \\ k^{−1}(U_{j} − U_{j−1}) + A_{h}U_{j} = P_{h}F (U_{j}), t_{j} > 0. \end{matrix} (59)

In a similar way as for the differential equation (4) one obtains a unique local solution of (59) for k ≤ k^{∗}. We denote its solution (for as long as it exists) by U_{j} = S_{hk}(t_{j})v_{h}. Again in a similar way as for the differential equation (4) one shows that for any bounded set B_{0} ⊂ H there is a bounded set B ⊂ H such that, for k ≤ k^{∗}(B_{0}),

S_{hk}(t_{n})(B_{0} ∩ H_{h}) ⊂ B, t_{n} ≥ 0. (60)

This means that the local solutions can be extended in time so that S_{hk}(t_{n}) exists for all t_{n} ≥ 0. Our main result is the following.

Theorem 3. Let B ⊂ H be a bounded set. There are positive numbers h^{∗}, k^{∗} and C such that for any h ≤ h^{∗}, k ≤ k^{∗} and v_{0} ∈ B ∩ H_{h} there exist T = T (h, k, B) and u ∈ H such that

|S_{hk}(t_{n})v_{0} − S(t_{n} − T)u| ≤ C(h + k), t_{n} ≥ T.

We need some preparations for the proof. From [16, Theorem 5.3] we deduce the following error bound. Recall that | · | denotes the norm of H_{0}^{1}; see (3).

Lemma 7. Let 0 < T_{1} < T_{2} and B_{0} ⊂ H be a bounded set. Assume that v ∈ B_{0}, v_{h} ∈ B_{0} ∩ H_{h}. Then, for k ≤ k^{∗}(B_{0}),

|S(t_{n})v − S_{hk}(t_{n})v_{h}| ≤ C(B_{0}, T_{1}, T_{2})(|v − v_{h}| + h + k), t_{n} ∈ [T_{1}, T_{2}]. (61)

Proof. Theorem 5.3 of [16] provides the following error bound. Let B ⊂ H be a bounded set. If S(t_{n})v ∈ B, S_{hk}(t_{n})v_{h} ∈ B for t_{n} ∈ [0, T_{2}], then for k ≤ k^{∗}(B),

|S(t_{n})v − S_{hk}(t_{n})v_{h}| ≤ C(B, T_{2})(|v − v_{h}| + ht_{n}^{−1/2} + kt_{n}^{−1}), t_{n} ∈ (0, T_{2}]. (62)

Restricting the time interval and using the a priori bounds (5) and (60) we obtain an error estimate of the desired form (61). The singularities t_{n}^{−1/2}, t_{n}^{−1} in (62) arise because the initial value v ∈ H is not sufficiently smooth for the error bound to hold uniformly as t_{n} → 0.

In the next lemma we show that the approximate solution moves into a neighborhood of the attractor A after a finite time and remains there.

Lemma 8. Let B ⊂ H be a bounded set and let ϵ > 0. There are positive numbers T_{0}, k^{∗}, h^{∗}, depending on ϵ and B, such that, for h ≤ h^{∗}, k ≤ k^{∗}, we have

dist(S_{hk}(t_{n})(B ∩ H_{h}), A) < ϵ, t_{n} ∈ [T_{0}, ∞).

Proof. See, e.g., the proof of [16, Theorem 4.5].

Proof of Theorem 3. Without loss of generality we assume that t = 1 is a mesh point and write σ(v) = S(1)v as before. Let d_{0}, L_{0}, W be given by Theorem 1. We shall extract from S_{hk}(t_{n})v_{0} a pseudotrajectory of σ for which we can apply Theorem 1.

Using Lemma 8 we find T_{0}, h^{∗}, k^{∗} depending on B and W such that

S_{hk}(t_{n})v_{0} ⊂ W, for t_{n} ≥ T_{0}, h ≤ h^{∗}, k ≤ k^{∗}.

We restrict h^{∗}, k^{∗} further so that

C(W, 1, 1)(h^{∗} + k^{∗}) ≤ d_{0};

cf. Lemma 7. For h ≤ h^{∗}, k ≤ k^{∗}, v_{0} ∈ B ∩ H_{h} we have

v′_{j} := S_{hk}(T_{0} + j)v_{0} ∈ W for j ≥ 0,

and, by Lemma 7,

|σ(v′_{j}) − v′_{j+1}| = |S(1)v′_{j} − S_{hk}(1)v′_{j}| ≤ C(W, 1, 1)(h + k),

so that v′_{j} is a d-pseudotrajectory of σ with d = C(W, 1, 1)(h + k) ≤ d_{0}. Lemma 3 gives N depending on d and W such that there exists m ∈ [0, N] with

dist(v′_{m}, M) ≤ 2d.

Take u′_{0} = S_{hk}(m)v′_{0} = S_{hk}(T_{0} + m)v_{0} and define u′_{j} = S_{hk}(j)u′_{0}. We have constructed u′_{j} so that it satisfies the assumptions of Theorem 1 and hence there is u′ ∈ H such that

|σ^{j}(u′) − u′_{j}| ≤ L_{0}d for j ≥ 0. (63)

We define T = T_{0} + N + 1 and

\begin{aligned}ũ & = S(N − m)u′, & ũ_{0} = S_{hk}(N − m)u′_{0}, \\ u & = S(1)ũ, & u_{0} = S_{hk}(1)ũ_{0}. \end{aligned}

Let l ≥ 0 and take t_{n} ∈ [T + l, T + l + 1]. Then

S_{hk}(t_{n})v_{0} = S_{hk}(t_{n} − T − l + 1)S_{hk}(l)ũ_{0} = S_{hk}(t_{n} − T − l + 1)u′_{j},
S(t_{n} − T)u = S(t_{n} − T − l + 1)S(l)ũ = S(t_{n} − T − l + 1)σ^{j}(u′),

where t_{n} − T − l + 1 ∈ [1, 2] and j = l + N − m. Lemma 7 and (63) then give

\begin{aligned}|S(t_{n} − T)u − S_{hk}(t_{n})v_{0}| & ≤ C(W, 1, 2)(|σ^{j}(u′) − u′_{j}| + h + k) \\ & ≤ C(W, 1, 2)(L_{0}d + h + k) ≤ C(h + k). \end{aligned}

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

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

Faculty of Mathematics and Mechanics, State University of S.-Petersburg, S.-Pet-ersburg, 198904, Russia

E-mail address: sp@spil.usr.pu.ru