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

Larsson, Stig; Sanz-Serna, J.-M.

doi:10.1090/S0025-5718-99-01038-8

A shadowing result with applications to finite element approximation of reaction-diffusion equations
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by Stig Larsson and J.-M. Sanz-Serna PDF

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

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.

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