Inverse inequalities on non-quasiuniform meshes and application to the Mortar element method

Wolfgang Dahmen, Birgit Faermann, Ivan G. Graham, Wolfgang Hackbusch, and Stefan A. Sauter

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Submission date: 19. May. 2001
Pages: 31
published in: Mathematics of computation, 73 (2004) 247, p. 1107-1138 
Bibtex
MSC-Numbers: 65N12, 65N30, 65N38, 65N55, 41A17, 46A35
Keywords and phrases: inequalities, mesh-dependent norms, inverse estimates, nonlinear approximation therory, non-matching grids, mortar element method, boundary element method
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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 locallyrefined shape-regular (but possibly non-quasiuniform) meshes. These inequalities involve norms of the form for positive and negative s and , 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 quasiuniform 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.