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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the existence of maximum principles in parabolic finite element equations
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by Vidar Thomée and Lars B. Wahlbin PDF
Math. Comp. 77 (2008), 11-19 Request permission

Abstract:

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.
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Additional Information
  • Vidar Thomée
  • Affiliation: Department of Mathematics, Chalmers University of Technology, S-41296 Göteborg, Sweden
  • MR Author ID: 172250
  • Email: thomee@math.chalmers.se
  • Lars B. Wahlbin
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
  • Email: wahlbin@math.cornell.edu
  • Received by editor(s): October 16, 2006
  • Received by editor(s) in revised form: November 10, 2006
  • Published electronically: May 14, 2007
  • Additional Notes: The authors were partly supported by the U.S. National Science Foundation under Grant DMS 0310539
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 77 (2008), 11-19
  • MSC (2000): Primary 65M12, 65M60
  • DOI: https://doi.org/10.1090/S0025-5718-07-02021-2
  • MathSciNet review: 2353941