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Backward Euler discretization of fully nonlinear parabolic problems


Authors: C. González, A. Ostermann, C. Palencia and M. Thalhammer
Journal: Math. Comp. 71 (2002), 125-145
MSC (2000): Primary 65M12, 65M15; Secondary 35K55, 35R35, 65L06, 65L20
DOI: https://doi.org/10.1090/S0025-5718-01-01330-8
Published electronically: July 22, 2001
MathSciNet review: 1862991
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Abstract:

This paper is concerned with the time discretization of nonlinear evolution equations. We work in an abstract Banach space setting of analytic semigroups that covers fully nonlinear parabolic initial-boundary value problems with smooth coefficients. We prove convergence of variable stepsize backward Euler discretizations under various smoothness assumptions on the exact solution. We further show that the geometric properties near a hyperbolic equilibrium are well captured by the discretization. A numerical example is given.


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Additional Information

C. González
Affiliation: Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, E-47011 Valladolid, Spain
Email: cesareo@mac.cie.uva.es

A. Ostermann
Affiliation: Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria
Address at time of publication: Section de mathématiques, Université de Genève, rue du Lièvre 2–4, CH-1211 Genève 24, Switzerland
Email: Alexander.Ostermann@uibk.ac.at, Alexander.Ostermann@math.unige.ch

C. Palencia
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Facultad de Ciencias, E-47011 Valladolid, Spain
Email: palencia@mac.cie.uva.es

M. Thalhammer
Affiliation: Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria
Email: Mechthild.Thalhammer@uibk.ac.at

DOI: https://doi.org/10.1090/S0025-5718-01-01330-8
Keywords: Nonlinear parabolic problems, time discretization, backward Euler method, convergence estimates, stability bounds, asymptotic stability, hyperbolic equilibrium.
Received by editor(s): January 6, 2000
Published electronically: July 22, 2001
Additional Notes: The authors acknowledge financial support from Acciones Integradas Hispano-Austríacas 1998/99
Article copyright: © Copyright 2001 American Mathematical Society

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