COMPUTING INERTIAL

. This paper discusses two numerical schemes that can be used to approximate inertial manifolds whose existence is given by one of the standard methods of proof. The methods considered are fully numerical, in that they take into account the need to interpolate the approximations of the manifold between a set of discrete gridpoints. As all the discretisations are reﬁned the approximations are shown to converge to the true manifold.

1. Introduction & summary.Since their introduction by Foias et al. [12], inertial manifolds have been an active area of research.These finite-dimensional, exponentially attracting, positively invariant manifolds offer a way of reducing an a priori infinite-dimensional partial differential equation to a finite set of ordinary differential equations, via an asymptotic "slaving rule" (cf.Haken [12]).
However, all the current existence results (see later for references) are nonconstructive, and hence it is of interest to develop methods that can provide more concrete representations of these manifolds.One approach has given rise to the theory of "approximate inertial manifolds" (Foias et al. [7], [10], [11]; Sell [29]), which seeks an explicit manifold that lies close to the global attractor (and is useful even when an exact inertial manifold is not known to exist).
In this paper we develop two computational methods for calculating to within an arbitrary degree of accuracy any inertial manifold whose existence can be proved when a standard spectral gap condition holds (see later).The advantage of such an approach is that one can theoretically relate the approximate manifold obtained to the dynamics of a particular discrete scheme, and (given sufficient computing time) the approximation can be refined with no extra analytical effort.The disadvantage is, as with any numerical scheme, the lack of a closed functional form for the approximation.
Note, however, the convergence of these approximations requires that a slightly strengthened form of the spectral gap condition holds.Since the most interesting questions in fact concern how inertial manifolds vary towards the limits of the standard gap condition (and beyond) these methods are clearly not ideal.However, it is hoped that they will serve as a first step towards more generally applicable methods.We discuss one possibility in the final section.
1.1.Preliminaries.Many interesting partial differential equations have a dissipativity property, meaning that all solutions will eventually enter a bounded set in the phase space.By truncating all the nonlinear terms to be zero outside this bounded absorbing set, the equation is rendered easier to analyse while preserving all its interesting asymptotic behaviour (see Foias et al. [9], Mallet-Paret & Sell [16], Robinson [27], or Temam [30]).
In such a way we arrive at the following general dissipative evolution equation on some Hilbert space H (with norm | • |), du/dt + Au + f (u) = 0. (1.1) Here A is a positive linear self-adjoint operator with a compact inverse: we denote by D(A α ) the domain of A α , and by | • | α the norm in this space, i.e.
We assume that the nonlinear term f • is globally bounded • and satisfies the global Lipschitz estimate For technical reasons we take 0 ≤ α ≤ 1/2, although this restriction can be relaxed (see Chow et al. [1] or Rodriguez Bernal [28]: in fact one can take f : In particular these conditions imply that corresponding to each initial condition u 0 ∈ D(A α ) there exists a unique solution of (1.1) -see Henry [13] and Miklavčič [18].
Since A is self-adjoint and A −1 is compact, A has a set of eigenvalues λ j and corresponding eigenfunctions w j , Aw j = λ j w j λ j+1 ≥ λ j (Renardy & Rogers [20]).The generalised Fourier basis {w j } ∞ j=1 gives rise to a sequence of finite-dimensional projection operators P N defined by For an N with λ N +1 = λ N we define the orthogonal complement of P N , Q N , All current existence results give the inertial manifold as the graph of some Lipschitz function φ : (1.5) An inertial manifold of this form is known to exist provided that there exist large enough gaps in the spectrum of the linear term A, i.e. if the spectral gap condition holds.This paper presents two numerical methods that can be used to approximate the function φ that gives the inertial manifold.There are three stages to the "fully numerical" method presented here.First the spatial dependence is discretised using a Galerkin truncation (section 2); then the time is made discrete (two simple timestepping methods are introduced in section 3); and finally some interpolation of φ (and its approximations along the way) is needed: one possible interpolation scheme is also introduced in section 3.
In section 4 a numerical scheme based on the Lyapunov-Perron fixed point method is shown to converge, while in section 5 we analyse a procedure based on the Hadamard graph transform.
2. The Galerkin truncation.The first step is to make the spatial dependence discrete by using a Galerkin truncation, which also has the happy effect of turning the original, infinite-dimensional, problem into a finite-dimensional one.
Setting all modes higher than N to zero gives Under the gap condition (1.6) these truncations possess inertial manifolds given as the graphs of functions φ N : R n → R N −n , One can show by various methods (Foias et al. [9]; Robinson [25]; Temam [30]) that φ N → φ in the C 0 topology as N → ∞.(In Jones & Stuart [14] and Jones et al. [15], the much stronger result of C 1 convergence of the manifolds is obtained.)Rather than write (2.1) repeatedly, in the remainder of this paper we consider the finite-dimensional system of ordinary differential equations on R N , where A is a positive symmetric N × N matrix, and f satisfies In this case the spectral gap condition (1.6) becomes (with α = 0) (2.3) Our task is to approximate φ N computationally.
3. Elements of the numerical approach.In this section we analyse two aspects of making the problem discrete.First we introduce two discrete time-stepping schemes, and prove that they converge as the timestep is refined.Secondly we consider notions of interpolation.We will use the notation F n l (often abbreviated F l when the value of n is clear) to denote globally bounded l-Lipschitz is approximated using a simple one-step method.We give a brief derivation of the methods based on the variation of constants formula, The idea is to retain the first term unchanged and to approximate the integrand by f (u(t)) or f (u(t + h)) to give 4) (a simple application of the contraction mapping theorem to the map J x = e −Ah y+ hf (x) shows that this method has is well-defined).The first of these two methods is akin to that of Demengel & Ghidaglia [4], who consider it is clear that these methods agree to first order in h.However, the two methods above are more suitable for what follows.
The following proposition shows that both these methods are accurate to O(h) on bounded time intervals.
where u • represents u k or u k+1 depending on whether the method is explicit or implicit. Thus, The norm of e As − 1 is bounded by e λ N s − 1 ≤ 2λ N h for sufficiently small h, and so Now all that is needed is a bound on the integrand, so that the change in u(t) between t = kh and t = (k + 1)h must be taken into account.
Now note that |u(t)| ≤ K 1 (u(0)) for all t ≥ 0: (using the variation of constants formula).It follows that and so certainly for the explicit scheme, and e −hλ1 /(1 − C 1 h) for the implicit scheme.In both cases Writing Λ = 2(C 1 − λ 1 )/3 and using the asymptotics of for h small enough.Thus, for kh ∈ [0, T ], where the final algebraic step follows since (1 + x/k) k → e x as k → ∞, and so (1 + x/k) k ≤ 2e x for k large enough, i.e. h small enough.

Interpolation.
One way in which this paper differs significantly from previous works is that in seeking a computational method it is necessary to take into account that it is only possible to deal with a discrete set of gridpoints over which the "manifold function" φ is defined.So issues of interpolating φ from these points will be important.
For some given set of gridpoints G ⊂ R n we introduce the discrete analogue of the space and define a norm on G by The interpolation scheme used in the analysis here is in fact obtained from an extension theorem due to McShane [17].Although easy to calculate, the interpolation coarsens the Lipschitz constant of the discrete function.Denoting the interpolation by φ → φ, it will be shown that where Lip(ψ) is the Lipschitz constant of ψ ∈ F l and Lip G (χ) is the "discrete Lipschitz constant" for χ ∈ G (i.e.l for χ ∈ G l ).In other words, Ideally the inequality would be an equality with σ = 1 (that such an "ideal" extension does exist is shown by Federer [5]).
In the statement of the proposition, Proof.Each component φ j (x) of φ(x) satisfies (3.6).It is straightforward to show (see [17]) that φj is an extension (and hence interpolant in this case) of φ j which also satisfies (3.6).
The convergence result follows since φ − [φ] G has Lipschitz constant at most 1 + σ ≤ 2σ, and any point in K lies within δ(G) of one of the points in G on which The following simple observation will be useful in what follows.If f and g have Lipschitz constants L f and L g respectively, then (3.7) 4. An approximation scheme based on the Lyapunov-Perron method.
We first use the Lyapunov-Perron fixed point method to provide a candidate approximation scheme.In what follows P = P n and Q = Q n .
In the continuous case the method involves finding the fixed point of an integral operator, and this corresponds to an invariant manifold for the equation (Chow et al. [1]; Foias et al. [9]; Henry [13]; Miklavčič [19]; Rodriguez-Bernal [28]; Temam [30]).For a given Lipschitz function φ ∈ F n and point p 0 ∈ R n denote by p(t) the solution of the equation the integral operator T , which maps φ into another function, is defined by Under the gap condition (2.3) T is a contraction mapping on F 1 , for some κ T < 1.
For the discrete time method used here, T has to be replaced by the sum where the p k satisfy the implicit relation from (3.4) with q k = φ(p k ) enforced for all k: with p 0 specified.Fixed points of T in F n will be invariant manifolds for the discrete implicit scheme the appropriate Q component of the solution on the invariant manifold.
The contraction mapping argument applied to T will show the existence of an invariant manifold.Note that this approach is similar to that of Demengel & Ghidaglia [4], except that the form of T is simplified by careful choice of the scheme (4.4).
The following simple lemma, on the growth of deviations in the successive backward iterates of the finite-dimensional p equation, is necessary in the proof of the proposition.
Lemma 4.1.The separation p 0 given, where ∆φ = sup Proof.We have and so Thus with where γ h is defined by and clearly has the behaviour given in the lemma.The estimate (4.5) now follows, since 4.1.The operator T as a contraction mapping.Lemma 4.1 is now applied in the analysis of T .
Proof.Clearly | T φ| is bounded by where the sum is bounded by the integral since the terms are decreasing.
To show that T φ is Lipschitz, consider Using the estimate (4.5) with ∆φ = 0 gives and provided λ n+1 > γ h the terms are decreasing and the sum can be bounded, as above, by an integral holds, T is a contraction mapping on F l , T φ − T φ ≤ κ T φ − φ (4.10) for some κ T < 1.It follows that the implicit scheme (3.4) has an invariant manifold given as the graph of some function φ ∈ F l .
Note that, using (4.7), the condition (4.9) is in fact a spectral gap condition once again, namely With l = 1 in the limit as h → 0 this reduces to the standard gap condition (2.3) Proof.First, it is clear from (4.8) that (4.9) implies that T maps F l into itself.To show that T is a contraction write Now using the estimate from (4.5) this gives and so for T to be a contraction the condition is Since the spectral gap condition (4.9) implies in particular that for h small enough, T is a contraction provided that The important term here is clearly the first, and it follows from the spectral gap condition that λ n+1 > (1 + l −1 )C 1 : thus T is a contraction provided that h is small enough (h < (1 + l)/(1 + l −1 ) certainly suffices).
Since T has a fixed point, the implicit scheme (3.4) has an invariant manifold given as the graph of some function Proof.Given > 0, first choose T so large that and (arguing similarly) that Then, since the simple integration scheme is O(h) for each fixed a provided that f is Lipschitz, and likewise the implicit scheme is O(h) on bounded time intervals, for any fixed T we have so choosing h small enough it follows that showing convergence as h → 0.
We note here that other more realistic schemes could be considered in a similar way -for example, a standard fourth order Runge-Kutta method is analysed in a similar way to section 3 in Robinson [22].

4.2.
Truncating the summation operator.The approximation to T given by the finite sum Tτ (τ = M h), can be computed numerically.It is clear that the finite sum Tτ approaches the infinite sum as the number of terms (i.e.τ ) increases.The existence of fixed points for Tτ and the convergence of these fixed points follows swiftly from the results of the previous subsection.
Proof.The argument of proposition 4.3, which showed that T is a contraction, can easily be adapted to treat Tτ and deduce the existence of fixed points φh,τ for each operator.Furthermore, the truncations of T , Tτ , converge uniformly to T on F: following the argument of proposition 4.4 we can easily obtain Now the convergence of the fixed points of Tτ to that of T as τ → ∞ follows simply (as above), since where κ T is the contraction constant of T from (4.10).Thus Thus Tτ is indeed a reasonable starting point for a numerical attempt to approximate φ.

Including interpolation.
We now have to take the problems arising from interpolation into account.Rather than having a function φ defined over the whole of P H, in fact we will have a function ψ defined only over a fixed grid Since the calculation of the backwards trajectory of p k will require values of φ at other points, we first have to replace ψ by its interpolation ψ, ψ ∈ F σ (see (3.5)).Once this is done we can apply the truncated summation operator Tτ introduced above: in theory this produces a new function Tτ ψ with Tτ ψ ∈ F σ (see (4.8)).Of course, in fact we will only compute the values of Tτ ψ over the particular gridpoints G, thus obtaining the somewhat notation-heavy expression (where [f ] G denotes the restriction of the function f to G as above) for the "fully numeric" version of T .We will denote this operator by T * , We expect that T * will map G 1 into G l .It remains to determine conditions that ensure that we can take l = 1 and to show that iterates of T * will produce an approximation to the fixed point of Tτ .This requires a slightly strengthened version of the spectral gap condition.
Proposition 4.6.Provided that then T * maps G 1 into itself.Furthermore, for any ψ ∈ G 1 , there exists an j 0 (ψ) such that for all j ≥ j 0 , where κ is the contraction constant of Tτ .
Note that as h → 0, (4.12) becomes Proof.To begin, note that if ψ ∈ G 1 then, from (3.5), ψ ∈ G σ .Now, we showed in proposition 4.2 that if φ ∈ F σ and the strengthened condition (4.12) holds then T φ (and also Tτ φ) is an element of F 1 .Thus T * maps G 1 into itself.
To obtain the approximation result (4.13), observe (with κ = κ Tτ and φ = φh,τ ) that since Tτ is a contraction on F σ when (4.12) holds.Thus and the iterates of ψ under T * satisfy After a sufficient number of iterations one has 5. An approximation scheme based on the graph transform method.It is also possible to produce an approximation scheme based on the more geometric "graph transform method" -this is the aim of this section.

5.1.
A discrete cone condition.The Hadamard or "graph transform" existence proof of Mallet-Paret & Sell [16] is based on following a particular set of initial conditions under the flow.Fundamental to this "evolution" method is the cone condition, which states that once the difference of two solutions lies in a certain cone in R N it remains there in the future (see Robinson [21] for a fuller discussion).This section gives a proof of a discrete version of this property for the one-step method.
We begin by investigating the evolution of Lipschitz graphs under the flow.
It is a consequence of this proposition that the image under the discrete evolution S h of a manifold given as the graph of an l-Lipschitz function φ is another Lipschitz manifold provided that e −λnh > (1 + l)C 1 h.In this case we define an evolution S h on the space of Lipschitz functions via From above, S h : F l → F l. Proposition 5.2.If the condition holds then S h maps F 1 into itself, and there is an inertial manifold given as the graph of (5.7) Note that (5.6) reduces to the spectral gap condition (2.3) as h → 0.
Proof.It is immediate from proposition 5.1 that if (5.6) holds then whenever for all k ≥ 0, and in particular it follows that S h maps F 1 into itself.As a first step to constructing an inertial manifold we note that so in particular if Qu 0 = 0 we have for all k ∈ Z + .

Now let
and so it follows from the "cone invariance" in (5.8) and (5.9) that the backwards iterates u −k and ū−k of u and ū must satisfy and so M = G[φ ∞ ] is an invariant manifold.That M attracts exponentially can either be shown following almost exactly the above argument as above, or more directly by noting that the invariance of G and thus, writing It is immediate that If this discrete scheme possesses inertial manifolds, they will converge to those of the continuous time system.This follows using the convergence result from Robinson [25] reproduced below (the notation is specialised to the case under consideration here).
Theorem 5.3.Suppose that a semigroup S(t) : R N → R N is approximated by a family of schemes {S h } h>0 , and that S h → S uniformly on bounded intervals of time and for initial conditions in bounded subsets of H. Then if each S h has an inertial manifold M h given as the graph of φ h ∈ F n l , and the rate of attraction is uniform, in that is an inertial manifold for S(t) that attracts at the same rate: (This theorem assumes nothing about the properties of the limiting equation; when one assumes that the limiting equation has an inertial manifold, Jones et al. [15] and Jones & Stuart [14] show that the manifolds for various approximation schemes converge to the "true" manifold in a C 1 fashion.) We now use the attraction property from (5.7) along with knowledge of how Lipschitz graphs behave under S h (5.2) to examine an approximation method based on the graph transform property.(5.11) We now want to ensure that (5.6) holds for every value of l with σ −1 ≤ l ≤ 1, so we assume that e −λn+1h + (1 + σ)C 1 h < e −λnh − 2C 1 h.
In particular this means that for every σ −1 ≤ l ≤ 1 we have l ≤ θl for some fixed θ < 1.Thus if φ ∈ F 1 we have Lip(S k h φ) ≤ θ k .In particular Lip(S d h φ) ≤ σ −1 for all φ ∈ F 1 if we take d sufficiently large.
We will write the interpolant of S d h ψ as E(S d h ψ) in order to emphasise that the interpolation is not over the regular grid G but over G .This gives a 1-Lipschitz function again, which we then restrict back to G, thus preventing the points that we are calculating with from separating too far.
We now show that if there is an invariant Lipschitz manifold that attracts exponentially then, provided that the grid is fine enough, this scheme will produce a good approximation.
The scheme can now be thought of as an operator T acting on 1-Lipschitz functions ψ ∈ G 1 , given by T φ = [E(S d h ψ)] G .
(5.12) (Note that there is some slight abuse of notation in the above definition, since strictly speaking S h acts on functions in F l rather than functions in G 1 .However, the procedure outlined above shows that in order to find the values of S d h φ over the transformed grid "S d h G" it is not actually necessary to interpolate over the initial gridpoints, but only over the transformed gridpoints.)holds.Then given ψ ∈ G σ , there exists an k 0 (ψ) such that, for all k ≥ k 0 , where φ ∞ is the function from proposition 5.2 and θ is defined in (5.14).

1 .
and supp(φ) ⊂ P n Ω ρ .Discrete time-stepping.In this section the time evolution of the equation du

Proposition 3 . 1 .
For both (3.3) and (3.4) there exists a constant K = K(T, X) such that |u(kh) − u k | ≤ Kh, for all 0 ≤ kh ≤ T , and all u 0 ∈ X, a bounded set in R N .I.e. the schemes converge uniformly on bounded time intervals and bounded sets in R N .Proof.The exact solution is given by u((k + 1)h) = e −Ah u(kh) − h 0 e −A(h−s) f (u(kh + s)) ds and the one-step method has

Proposition 3 . 2 .
Let G be a discrete set of points contained within a compact subset K of R n , and φ : G → R m a Lipschitz continuous function such that|φ(x) − φ(y)| ≤ C|x − y| x, y ∈ G. (3.6)Then the function φ, with componentsφj (x) = sup y∈G φ j (y) − C|x − y| ,is an interpolant of φ and (3.5) holds with σ = √ m.Furthermore the interpolation scheme converges as the grid is refined, in that ifs(G) ≡ sup φ∈F [φ] G − φ K ,where [f ] G denotes the restriction of the function f to G, and δ(G) = sup x∈K inf g∈G |x − g| then s(G) ≤ 2σδ(G).

5. 2 .
Including interpolation.If initially one considers only the graph of ψ ∈ G 1 over the discrete set of gridpoints in G, under the evolution this will become the setG h ≡ {S h (g + ψ(g)) : g ∈ G.}In this case, in something of an abuse of notation, we will also defineS h G = P G h ,i.e. the "evolved" grid.The numerical scheme begins with a 1-Lipschitz function ψ defined over the discrete grid G, ψ ∈ G 1There is no need to interpolate at this stage.The collection of initial points {g + ψ(g)} are now evolved d times (d will be specified below) to give a new set of points{S d h (g + ψ(g)) : g ∈ G}(5.10) that lie on G[S d h ψ].These points give values of the function S d h ψ, defined over the new grid G = P {S d h (g + ψ(g)) : g ∈ G}.Note that |P w k+1 | ≤ (e −λ1h + 2hC 1 )|P w k |, and so δ(G ) ≤ (e −λ1h + 2hC 1 ) d δ(G).