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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2D
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by James H. Adler and Victor Nistor PDF
Math. Comp. 84 (2015), 2191-2220 Request permission


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:
  • 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:
  • 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)
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 2191-2220
  • MSC (2010): Primary 65N30; Secondary 65N50
  • DOI:
  • MathSciNet review: 3356024