The authors consider the evolution equation \[ {du\over dt}+ Au= F(u),\quad u(0)= u_0\tag{1} \] in a separable Hilbert space \(X\) with inner product \((\cdot, \cdot)\). \(A\) is assumed to be a densely defined sectorial operator with compact inverse and eigenfunctions \(\{\varphi_i\}\). Such operator \(A\) generates an analytic semigroup \(L(t)\) for \(t\geq 0\). \(F\) is assumed to satisfy sufficient conditions so that (1) generates a semigroup \(S(t): X^\gamma\to X^\gamma\) for some \(\gamma\geq 0\) and for each \(t\geq 0\). Since their aim is to study the equation (1) under approximation and since standard error estimates for numerical schemes are formulated over finite time intervals, they find it convenient to write the solutions of (1) as solutions of the time \(T\) map of the flow.
The solution of (1) at time \(t\) can be written as \[ u(t)= L(t) u_0+ N(u_0, t),\text{ where } L(t):= e^{- At},\quad N(\nu, t):= \int^t_0 L(t- s) F(S(s)\nu)ds. \] The sequence \(u_m:= u(mT)\) then satisfies \[ u_{m+ 1}= G(u_m),\text{ where } G(u):= Lu+ N(u),\quad L:= L(t),\quad N(\nu):= N(\nu, T). \] To construct an attractive invariant manifold they decompose the space \(X\) as follows: Let \(P\) denote the spectral projection associated with \(\{\varphi_1,\dots, \varphi_q\}\), the first \(q\) eigenfunctions of \(A\), and \(Q\)=\(I\)-\(P\). Let \(Y\)=\(PX\) and \(Z\)=\(QX\), so we have a decomposition of the space \(X\)=\(Y\oplus Z\). Assuming that the approximation of \(A\) yields a sectorial operator \(A^h\) with eigenfunctions \(\{\varphi^h_m\}\), and denoting by \(u^h_m\) the approximation to \(u_m\) which lies in some space \(V^h\subset X^\gamma\), \(E^h: X\to V^h\) the projection onto the approximating subspace, \(P^h: X\to Y^h\) the projection onto the first \(m\) eigenfunctions of \(A^h\), \(Q^h: X\to Z^h\), \(Q^h= E^h- P^h\), where \(V^h= Y^h\oplus Z^h\), \(Y^h= P^h X\), \(Z^h= Q^h X\) the approximation process then yields a mapping \(u^h_{m+ 1}= G^h(u^h_m)\).
The main theorem of this paper, concerning the relationship between the attractive invariant manifolds of the true and approximate map, is as follows: Under some assumptions on \(G\) and \(G^h\), both mappings possess attractive invariant manifolds representable as graphs of \(\Phi: Y\to Z\) and \(\Phi^h: Y^h\to Z^h\), respectively.