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Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D


Authors: James H. Adler and Victor Nistor
Journal: Math. Comp. 84 (2015), 2191-2220
MSC (2010): Primary 65N30; Secondary 65N50
DOI: https://doi.org/10.1090/S0025-5718-2015-02934-2
Published electronically: February 26, 2015
MathSciNet review: 3356024
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Abstract: We study the approximation properties of some general finite-element spaces constructed using improved graded meshes. In our results, either the approximating function or the function to be approximated (or both) are in a weighted Sobolev space. We consider also the $ L^p$-version of these spaces. The finite-element spaces that we define are obtained from conformally invariant families of finite elements (no affine invariance is used), stressing the use of elements that lead to higher regularity finite-element spaces. We prove that for a suitable grading of the meshes, one obtains the usual optimal approximation results. We provide a construction of these spaces that does not lead to long, ``skinny'' triangles. Our results are then used to obtain $ L^2$-error estimates and $ h^m$-quasi-optimal rates of convergence for the FEM approximation of solutions of strongly elliptic interface/boundary value problems.


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Additional Information

James H. Adler
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
Email: james.adler@tufts.edu

Victor Nistor
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 – and – Université de Lorraine, UFR MIM, Ile du Saulcy, CS 50128, 57045 METZ, France
Email: nistor@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02934-2
Keywords: Finite elements, singularities, graded meshes
Received by editor(s): September 22, 2012
Received by editor(s) in revised form: September 13, 2013, and December 22, 2013
Published electronically: February 26, 2015
Additional Notes: The second author was partially supported by NSF Grants OCI-0749202, DMS-1016556 and ANR-14-CE25-0012-01 (SINGSTAR)
Article copyright: © Copyright 2015 American Mathematical Society