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Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Authors: W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch and S. A. Sauter
Journal: Math. Comp. 73 (2004), 1107-1138
MSC (2000): Primary 65N12, 65N30, 65N38, 65N55, 41A17, 46E35
Published electronically: July 14, 2003
MathSciNet review: 2047080
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Abstract: 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

B. Faermann
Affiliation: Institut für Mathematik, Sekretariat MA 4-5, Technische Univerität Berlin, D-10623 Berlin, Germany

I. G. Graham
Affiliation: Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
MR Author ID: 76020

W. Hackbusch
Affiliation: Max-Planck-Institut Mathematik in den Naturwissenschaften, D-04103 Leipzig, Inselstr. 22-26, Germany

S. A. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland
MR Author ID: 313335

Keywords: Inequalities, mesh-dependent norms, inverse estimates, nonlinear approximation theory, nonmatching grids, mortar element method, boundary element method
Received by editor(s): May 2, 2001
Received by editor(s) in revised form: January 10, 2003
Published electronically: July 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society