A shadowing result with applications to finite element approximation of reaction-diffusion equations
Stig Larsson, Jesús María Sanz-Serna · Mathematics of Computation · 1999
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 L_2 and H 1 H^1 norms hold uniformly over arbitrarily long time intervals.