Numerical Shadowing near the Global Attractor for a Semilinear Parabolic Equation

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 nite-dimensional systems. When applied to a semilinear parabolic problem in one space variable, approximated by a standard nite 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.