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A shadowing result with applications to finite element approximation of reaction-diffusion equations
Author(s):
Stig
Larsson;
J.-M.
Sanz-Serna.
Journal:
Math. Comp.
68
(1999),
55-72.
MSC (1991):
Primary 65M15, 65M60
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  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:
10.1090/S0025-5718-99-01038-8
PII:
S 0025-5718(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.
Copyright of article:
Copyright
1999,
American Mathematical Society
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