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

A shadowing result with applications to finite element
approximation of reaction-diffusion equations,

Math. Comp. 68 (1999), 55-72. 

Subjclass: 65M15, 65M60

Keywords: Shadowing, semilinear parabolic problem, hyperbolic
  stationary point, finite element method, backward Euler, error
  estimate

Abstract:  A shadowing result is formulated in such a way that
  it applies in the context of numerical approximations of semilinear
  parabolic problems.  The qualitative behavior of temporally and
  spatially discrete finite element solutions of a reaction-diffusion
  system near a hyperbolic equilibrium is then 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 the equilibrium.  Error bounds of
  optimal order in the $L_2$ and $H^1$ norms hold uniformly over
  arbitrarily long time intervals.

Address: Stig Larsson, 
  Department of Mathematics, 
  Chalmers University of Technology and G\"oteborg University, 
  SE--412 96 G\"oteborg, Sweden

E-mail: stig@math.chalmers.se

Address: J.-M.~Sanz-Serna, 
  Departamento de Matem\'atica Aplicada y Computaci\'on, 
  Facultad de Ciencias, Universidad de Valladolid,
  Valladolid, Spain

E-mail: sanzserna@cpd.uva.es