Implicit-explicit multistep finite element methods for nonlinear parabolic problems

Akrivis, Georgios; Crouzeix, Michel; Makridakis, Charalambos

doi:10.1090/S0025-5718-98-00930-2

Implicit-explicit multistep finite element methods for nonlinear parabolic problems

Authors: Georgios Akrivis, Michel Crouzeix and Charalambos Makridakis
Journal: Math. Comp. 67 (1998), 457-477
MSC (1991): Primary 65M60, 65M12; Secondary 65L06
DOI: https://doi.org/10.1090/S0025-5718-98-00930-2
MathSciNet review: 1458216
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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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