On the existence of maximum principles in parabolic finite element equations

Authors:
Vidar Thomée and Lars B. Wahlbin

Journal:
Math. Comp. **77** (2008), 11-19

MSC (2000):
Primary 65M12, 65M60

Published electronically:
May 14, 2007

MathSciNet review:
2353941

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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.

**1.**Andrei Drăgănescu, Todd F. Dupont, and L. Ridgway Scott,*Failure of the discrete maximum principle for an elliptic finite element problem*, Math. Comp.**74**(2005), no. 249, 1–23 (electronic). MR**2085400**, 10.1090/S0025-5718-04-01651-5**2.**H. Fujii,*Some remarks on finite element analysis of time-dependent field problems*, in Theory and Practice in Finite Element Structural Analysis, University of Tokyo Press, Tokyo, 1973, pp. 91-106.**3.**Sergey Korotov, Michal Křížek, and Pekka Neittaanmäki,*Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle*, Math. Comp.**70**(2001), no. 233, 107–119 (electronic). MR**1803125**, 10.1090/S0025-5718-00-01270-9**4.**Vitoriano Ruas Santos,*On the strong maximum principle for some piecewise linear finite element approximate problems of nonpositive type*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**29**(1982), no. 2, 473–491. MR**672072****5.**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377****6.**Vidar Thomée,*Galerkin finite element methods for parabolic problems*, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR**2249024****7.**Jinchao Xu and Ludmil Zikatanov,*A monotone finite element scheme for convection-diffusion equations*, Math. Comp.**68**(1999), no. 228, 1429–1446. MR**1654022**, 10.1090/S0025-5718-99-01148-5

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Additional Information

**Vidar Thomée**

Affiliation:
Department of Mathematics, Chalmers University of Technology, S-41296 Göteborg, Sweden

Email:
thomee@math.chalmers.se

**Lars B. Wahlbin**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
wahlbin@math.cornell.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-07-02021-2

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

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

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.