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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  and
Jean-Luc Guermond
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 $\ell^p$ -conditionnumber of the stiffness matrix resulting from the finite elementapproximation of a linear, abstract model problem. Sharp estimates interms of the meshsize h are obtained. The theoretical results areapplied to finite element approximations of elliptic PDE's invariational and in mixed form, and to first-order PDE's approximatedusing the Galerkin–Least Squares technique or bymeans of a non-standard Galerkin technique in L 1(Ω). Numerical simulations are presented to illustrate thetheoretical results.

Keywords

Finite elementscondition numberpartial differentialequationslinear algebra.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 1 , January 2006 , pp. 29 - 48
Copyright
© EDP Sciences, SMAI, 2006

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