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Mathematics of Computation

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Inverse-type estimates on $ hp$-finite element spaces and applications

Author: Emmanuil H. Georgoulis
Journal: Math. Comp. 77 (2008), 201-219
MSC (2000): Primary 65J05; Secondary 65R20
Published electronically: September 18, 2007
MathSciNet review: 2353949
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Abstract: This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to $ hp$-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The assumptions on the $ hp$-finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of $ hp$-boundary element method discretisations of integral equations, with element-wise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.

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Additional Information

Emmanuil H. Georgoulis
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

Keywords: Inverse-type inequalities, $hp$-finite element spaces, condition number, boundary element methods
Received by editor(s): February 27, 2006
Received by editor(s) in revised form: December 4, 2006
Published electronically: September 18, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.