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Failure of the discrete maximum principle for an elliptic finite element problem

Authors: Andrei Drăgănescu, Todd F. Dupont and L. Ridgway Scott
Journal: Math. Comp. 74 (2005), 1-23
MSC (2000): Primary 65N30; Secondary 65N50
Published electronically: March 23, 2004
MathSciNet review: 2085400
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Abstract: There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green’s function is positive for triangular meshes allowing sufficiently good approximation of $H^1$ functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green’s functions that are of independent interest.

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Additional Information

Andrei Drăgănescu
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Todd F. Dupont
Affiliation: Department of Computer Science, University of Chicago, Chicago, Illinois 60637

L. Ridgway Scott
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
MR Author ID: 157720

Keywords: Finite elements, Green’s function, discrete maximum principle
Received by editor(s): May 12, 2003
Received by editor(s) in revised form: June 8, 2003
Published electronically: March 23, 2004
Additional Notes: The work of the authors was supported by the ASCI Flash Center at the University of Chicago under DOE contract B532820 and by the MRSEC Program of the National Science Foundation under award DMR-0213745
Article copyright: © Copyright 2004 American Mathematical Society