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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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Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method
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by W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch and S. A. Sauter PDF
Math. Comp. 73 (2004), 1107-1138 Request permission


We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions $u$ defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form $\left \|h^{\alpha }u\right \| _{W^{s,p}(\Omega )}$ for positive and negative $s$ and $\alpha$, where $h$ is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is $N$, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results—previously known only for quasi-uniform meshes—to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.
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Additional Information
  • W. Dahmen
  • Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52062 Aachen, Germany
  • MR Author ID: 54100
  • Email:
  • B. Faermann
  • Affiliation: Institut für Mathematik, Sekretariat MA 4-5, Technische Univerität Berlin, D-10623 Berlin, Germany
  • Email:
  • I. G. Graham
  • Affiliation: Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
  • MR Author ID: 76020
  • Email:
  • W. Hackbusch
  • Affiliation: Max-Planck-Institut Mathematik in den Naturwissenschaften, D-04103 Leipzig, Inselstr. 22-26, Germany
  • Email:
  • S. A. Sauter
  • Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland
  • MR Author ID: 313335
  • Email:
  • Received by editor(s): May 2, 2001
  • Received by editor(s) in revised form: January 10, 2003
  • Published electronically: July 14, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Math. Comp. 73 (2004), 1107-1138
  • MSC (2000): Primary 65N12, 65N30, 65N38, 65N55, 41A17, 46E35
  • DOI:
  • MathSciNet review: 2047080