Evaluation of the condition number in linear systems arising in finite element approximations

Ern, Alexandre; Guermond, Jean-Luc

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 29-48.

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 $L^1\left(\Omega \right)$. Numerical simulations are presented to illustrate the theoretical results.

Classification : 65F35, 65N30
Mots clés : finite elements, condition number, partial differential equations, linear algebra

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