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Alternating direction finite volume element methods for 2D parabolic partial differential equations. (English) Zbl 1135.65037

The author combines finite volume methods and alternating direction methods for two dimensional parabolic equations. He adopts the ideas of J. Douglas jun. and co-authors [Numerical Solution partial diff. Equations II, Proc. 2nd Sympos. numerical Solution partial diff. Equations, SYNSPADE 1970, Univ. Maryland, 133–214 (1971; Zbl 0239.65088) and Math. Models Methods Appl. Sci. 11, No. 9, 1563–1579 (2001; Zbl 1012.65095)] and writes the finite volume element schemes as tensor product forms so that he can convert them to a series of one-dimensional problems, which can be solved alternatively.
He gives three kinds of alternating direction methods, the first two are similar to Douglas schemes [Zbl 0239.65088] and [Zbl 1012.65095] in the finite element method and the finite difference method, the third is an extension of the locally one-dimensional finite difference scheme [Zbl 1012.65095] with second order accuracy. He obtains optimal error estimates in \(L_2\) or \(H^1\) semi-norms for these schemes and illustrates that in two numerical examples.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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