ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

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

Ern, Alexandrea1 and Guermond, Jean-Luca2

a1 CERMICS, École nationale des ponts et chaussées, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France. ern@cermics.enpc.fr

a2 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

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 L 1(Ω). Numerical simulations are presented to illustrate the theoretical results.

(Received March 7 2005)

(Revised July 6 2005)

(Online publication February 23 2006)

Key Words:

  • Finite elements;
  • condition number;
  • partial differential equations;
  • linear algebra.

Mathematics Subject Classification:

  • 65F35;
  • 65N30
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