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Sharply local pointwise a posteriori error estimates for parabolic problems


Authors: Alan Demlow and Charalambos Makridakis
Journal: Math. Comp. 79 (2010), 1233-1262
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-10-02346-X
Published electronically: March 1, 2010
MathSciNet review: 2629992
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Abstract: We prove pointwise a posteriori error estimates for semi- and fully-discrete finite element methods for approximating the solution $ u$ to a parabolic model problem. Our estimates may be used to bound the finite element error $ \Vert u-u_h\Vert _{L_\infty(D)}$, where $ D$ is an arbitrary subset of the space-time domain of the definition of the given PDE. In contrast to standard global error estimates, these estimators de-emphasize spatial error contributions from space-time regions removed from $ D$. Our results are valid on arbitrary shape-regular simplicial meshes which may change in time, and also provide insight into the contribution of mesh change to local errors. When implemented in an adaptive method, these estimates require only enough spatial mesh refinement away from $ D$ in order to ensure that local solution quality is not polluted by global effects.


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Additional Information

Alan Demlow
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506–0027
Email: demlow@ms.uky.edu

Charalambos Makridakis
Affiliation: Department of Applied Mathematics, University of Crete, GR-71409 Heraklion, Greece; and Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Vasilika Vouton P.O. Box 1527, GR-71110 Heraklion, Greece
Email: makr@tem.uoc.gr

DOI: https://doi.org/10.1090/S0025-5718-10-02346-X
Keywords: Parabolic partial differential equations, finite element methods, adaptive methods, a posteriori error estimates, pointwise error estimates, maximum norm error estimates, localized error estimates, local error estimates
Received by editor(s): November 13, 2007
Received by editor(s) in revised form: April 26, 2009, and July 22, 2009
Published electronically: March 1, 2010
Additional Notes: The first author was supported in part by National Science Foundation grant DMS-0713770.
Article copyright: © Copyright 2010 American Mathematical Society