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


Dahmen, W; Faermann, B; Graham, I; Hackbusch, W; Sauter, S (2004). Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method. Mathematics of Computation, 73(247):1107-1138 (electronic).

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 ∥h α u∥ W s,p (Ω) for positive and negative s and α, 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.

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 ∥h α u∥ W s,p (Ω) for positive and negative s and α, 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 indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Inequalities, mesh-dependent norms, inverse estimates, nonlinear approximation theory, nonmatching grids, mortar element method, boundary element method
Language:English
Date:2004
Deposited On:29 Nov 2010 16:26
Last Modified:05 Apr 2016 13:24
Publisher:American Mathematical Society
ISSN:0025-5718
Additional Information:First published in [Math. Comp. 73 (2004), no. 247], published by the American Mathematical Society
Publisher DOI:https://doi.org/10.1090/S0025-5718-03-01583-7
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2047080
Permanent URL: https://doi.org/10.5167/uzh-21787

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