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On the existence of maximum principles in parabolic finite element equations

Vidar Thomée (Institutionen för matematiska vetenskaper, matematik) ; L. B. Wahlbin
Math. Comp. (0025-5718 ). Vol. 77 (2008), 261, p. 11-19.
[Artikel, refereegranskad vetenskaplig]

In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of Delauney type. In this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem. We then show that for the lumped mass method the above conditions on the triangulation are essentially sharp. This is in contrast to the elliptic case in which the requirements are weaker. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction or a positive operator.

Nyckelord: Maximum principle, parabolic equations, finite elements, lumped mass


DOI = 10.1090/S0025-5718-07-02021-2



Denna post skapades 2008-12-05. Senast ändrad 2010-01-26.
CPL Pubid: 79818

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Tillämpad matematik
Numerisk analys

Chalmers infrastruktur