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Evaluation of the condition number in linear systems arising in finiteelement approximations

Published online by Cambridge University Press:  23 February 2006

Alexandre Ern
Affiliation:
CERMICS, École nationale des ponts et chaussées, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France. ern@cermics.enpc.fr
Jean-Luc Guermond
Affiliation:
Dept. Math, Texas A&M, College Station, TX 77843-3368, USA and LIMSI (CNRS-UPR 3152), BP 133, 91403, Orsay, France. guermond@math.tamu.edu
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Abstract

This paper derives upper and lower bounds for the $\ell^p$-condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize h are obtained. The theoretical results are applied to finite element approximations of elliptic PDE's in variational and in mixed form, and to first-order PDE's approximated using the Galerkin–Least Squares technique or by means of a non-standard Galerkin technique in L1(Ω). Numerical simulations are presented to illustrate the theoretical results.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

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