Mathematics of Computation

Implicit-explicit multistep finite element methods for nonlinear parabolic problems

Author(s): Georgios Akrivis; Michel Crouzeix; Charalambos Makridakis.
Journal: Math. Comp. 67 (1998), 457-477.
MSC (1991): Primary 65M60, 65M12; Secondary 65L06
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Abstract: We approximate the solution of initial boundary value problems for nonlinear parabolic equations. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient, since their implementation requires at each time step the solution of a linear system with the same matrix for all time levels. We derive optimal order error estimates. The abstract results are applied to the Kuramoto-Sivashinsky and the Cahn-Hilliard equations in one dimension, as well as to a class of reaction diffusion equations in ${\mathbb{R}} ^{\nu },$ $\nu = 2, 3.$

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Retrieve articles in Mathematics of Computation with MSC (1991): 65M60, 65M12, 65L06

Georgios Akrivis
Affiliation: Department of Computer Science, University of Ioannina, 451 10 Ioannina, Greece
Email: akrivis@cs.uoi.gr

Michel Crouzeix
Affiliation: IRMAR, Université de Rennes I, Campus de Beaulieu, F-35042 Rennes, France
Email: michel.crouzeix@univ-rennes1.fr

Charalambos Makridakis
Affiliation: Department of Mathematics, University of Crete, 714 09 Heraklion, Crete, Greece, and IACM, Foundation for Research and Technology - Hellas, 711 10 Heraklion, Crete, Greece
Email: makr@sargos.math.uch.gr

DOI: 10.1090/S0025-5718-98-00930-2
PII: S 0025-5718(98)00930-2
Received by editor(s): July 3, 1995
Received by editor(s) in revised form: December 8, 1995
Additional Notes: The work of the first and third authors was supported in part by a research grant from the University of Crete
Copyright of article: Copyright 1998, American Mathematical Society

J. Frank, W. Hundsdorfer, J. W. Verwer, On the stability of implicit-explicit linear multistep methods, Appl. Numer. Math. 25 (1997), 193--205.

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P. K. Moore, An incomplete assemply with thresholding algorithm for systems of reaction-diffusion equations in three space dimensions IAT for reaction-diffusion systems, J. Comp. Phys. 189 (2003), 130-158.

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V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, first, Springer-Verlag, Berlin, 1997. (English)

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