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Localized pointwise error estimates for mixed finite element methods

Author: Alan Demlow
Journal: Math. Comp. 73 (2004), 1623-1653
MSC (2000): Primary 65N30, 65N15
Published electronically: March 23, 2004
MathSciNet review: 2059729
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Abstract: In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-Douglas-Marini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.

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Additional Information

Alan Demlow
Affiliation: Abteilung für Angewandte Mathematik, Hermann-Herder-Str. 10, 79104 Freiburg, Germany
MR Author ID: 693541

Keywords: Mixed finite element methods, pointwise error estimates
Received by editor(s): May 22, 2002
Published electronically: March 23, 2004
Additional Notes: This material is based on work supported under a National Science Foundation graduate fellowship and under NSF grant DMS-0071412
Article copyright: © Copyright 2004 American Mathematical Society