Evaluation of the condition number in linear systems arising in finite element approximations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, p. 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.

DOI : https://doi.org/10.1051/m2an:2006006
Classification:  65F35,  65N30
Keywords: finite elements, condition number, partial differential equations, linear algebra
@article{M2AN_2006__40_1_29_0,
author = {Ern, Alexandre and Guermond, Jean-Luc},
title = {Evaluation of the condition number in linear systems arising in finite element approximations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {1},
year = {2006},
pages = {29-48},
doi = {10.1051/m2an:2006006},
zbl = {pre05038391},
mrnumber = {2223503},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_1_29_0}
}

Ern, Alexandre; Guermond, Jean-Luc. Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, pp. 29-48. doi : 10.1051/m2an:2006006. http://www.numdam.org/item/M2AN_2006__40_1_29_0/

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