The Behavior of Finite Element Solutions of Semilinear Parabolic Problems Near Stationary Points

Abstract

The qualitative behavior of spatially semidiscrete finite element solutions of a semilinear parabolic problem near an unstable hyperbolic equilibrium $\overline u $ is studied. It is shown that any continuous trajectory is approximated by an appropriate discrete trajectory, and vice versa, as long as they remain in a sufficiently small neighborhood of $\overline u $. Error bounds of optimal order in the $L_2 $ and $H^1 $ norms hold uniformly over arbitrarily long time intervals. In particular, the local stable and unstable manifolds of the discrete problem converge to their continuous counterparts. Therefore, the discretized dynamical system has the same qualitative behavior near $\overline u $ as the continuous system.