S. Larsson and J.-M. Sanz-Serna

The behavior of finite element solutions of 
semilinear parabolic problems near stationary points

SIAM J. Numer. Anal. 31 (1994), 1000-1018

Abstract: 
We study the qualitative behavior of spatially semidiscrete
finite element solutions of a semilinear parabolic problem near an
unstable hyperbolic equilibrium $\ubar$.  We show 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 $\ubar$.  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
$\ubar$ as the continuous system.

Subjclass: 65M15, 65M60 

Keywords: Semilinear parabolic problem, unstable hyperbolic stationary
point, finite element method, error estimate, stable, unstable
manifold, shadowing