Finite elements on degenerate meshes: inverse-type inequalities and applications

Ivan G. Graham, Wolfgang Hackbusch, and Stefan A. Sauter

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Submission date: 27. Nov. 2002
Pages: 29
published in: IMA journal of numerical analysis, 25 (2005) 2, p. 379-407 
DOI number (of the published article): 10.1093/imanum/drh017
Bibtex
Keywords and phrases: finite elements, degenerate meshes, graded meshes
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Abstract:
In this paper we obtain a range of inverse-type inequalities which are applicable to finite element functions on general classes of meshes, including degenerate meshes obtained by anisotropic refinement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. We give two applications of these estimates in the context of boundary elements: (i) to the analysis of quadrature error in discrete Galerkin methods and (ii) to the analysis of the panel clustering algorithm. Our results show that degeneracy in the meshes yields no degradation in the approximation properties of these methods.