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Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems

Author: Alan Demlow
Journal: Math. Comp. 76 (2007), 19-42
MSC (2000): Primary 65N30, 65N15
Published electronically: October 4, 2006
MathSciNet review: 2261010
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Abstract: We prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second-order linear elliptic model problem. First we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error (the ``pollution term''). Using a mesh-dependent weight, the residual term is bounded in a sharply localized fashion. In specific situations the pollution term may also be bounded by computable residual estimators. On nonconvex polygonal and polyhedral domains in two and three space dimensions, we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks. The finite element mesh is only required to be simplicial and shape-regular, so that highly graded and unstructured meshes are allowed.

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

Keywords: Finite element methods, elliptic problems, a posteriori error estimation, local error analysis, pointwise error analysis
Received by editor(s): December 10, 2004
Received by editor(s) in revised form: September 16, 2005
Published electronically: October 4, 2006
Additional Notes: This material is based upon work partially supported under a National Science Foundation postdoctoral research fellowship.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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