A shadowing result with applications

to finite element approximation

of reaction-diffusion equations

Authors:
Stig Larsson and J.-M. Sanz-Serna

Journal:
Math. Comp. **68** (1999), 55-72

MSC (1991):
Primary 65M15, 65M60

DOI:
https://doi.org/10.1090/S0025-5718-99-01038-8

MathSciNet review:
1620227

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Abstract | References | Similar Articles | Additional Information

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 and norms hold uniformly over arbitrarily long time intervals.

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Additional Information

**Stig Larsson**

Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S–412 96 Göteborg, Sweden

Email:
stig@math.chalmers.se

**J.-M. Sanz-Serna**

Affiliation:
Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

Email:
sanzserna@cpd.uva.es

DOI:
https://doi.org/10.1090/S0025-5718-99-01038-8

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

Received by editor(s):
February 6, 1996

Additional Notes:
The first author was partly supported by the Swedish Research Council for Engineering Sciences (TFR). The second author was partly supported by “Dirección General de Investigación Científica y Técnica” of Spain under project PB92–254.

Article copyright:
© Copyright 1999
American Mathematical Society