Stig Larsson and Sergei Yu. Pilyugin,

Numerical shadowing near the global attractor 
for a semilinear parabolic equation,

preprint 1998:21, 
Department of Mathematics, Chalmers University of Technology.

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

Keywords. Semilinear, parabolic, Lipschitz shadowing property,
  attractor, Morse-Smale, inertial manifold, numerical method, finite
  element, backward Euler, error estimate, long-time.