Approximation of dissipative partial differential equations over long time intervals

Abstract

In this article the numerical analysis of dissipative semilinear evolution equations with sectorial linear part is reviewed. In particular the approximation theory for such equations over long time intervals is discussed. Emphasis is placed on studying the effect of approximation on certain invariant objects which play an important role in understanding long time dynamics. Specifically the existence of absorbing sets, the upper and lower semicontinuity of global attractors and the existence and convergence of attractive invariant manifolds, such as the inertial manifold and unstable manifolds of equilibrium points, is studied.