Attractive Invariant Manifolds under Approximation Part I: Inertial Manifolds

A class of nonlinear dissipative partial diierential equations that possess nite dimensional attractive invariant manifolds is considered. An existence and perturbation theory is developed which uniies the cases of unstable manifolds and inertial mani-folds into a single framework. It is shown that certain approximations of these equations , such as those arising from spectral or nite element methods in space, one-step time-discretization or a combination of both, also have attractive invariant manifolds. Convergence of the approximate manifolds to the true manifolds is established as the approximation is reened. In this part of the paper applications to the behavior of in-ertial manifolds under approximation are considered. From this analysis deductions about the structure of the attractor and the ow on the attractor under discretization can be made.