Best approximation property in the 𝑊¹_{∞} norm for finite element methods on graded meshes

Demlow, A.; Leykekhman, D.; Schatz, A.; Wahlbin, L.

doi:10.1090/S0025-5718-2011-02546-9

Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes
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by A. Demlow, D. Leykekhman, A. H. Schatz and L. B. Wahlbin;

Math. Comp. 81 (2012), 743-764

DOI: https://doi.org/10.1090/S0025-5718-2011-02546-9

Published electronically: September 29, 2011

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Abstract:

We consider finite element methods for a model second-order elliptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the $W^1_{\infty }$ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof technique. This result holds under a condition on the grid which is mildly more restrictive than the shape regularity condition typically enforced in adaptive codes. The second main contribution of this work is a discussion of the properties of and relationships between similar mesh restrictions that have appeared in the literature.

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