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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: 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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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