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

Alexandre Ern; Jean-Luc Guermond

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 1, page 29-48
  • ISSN: 0764-583X

Abstract

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This paper derives upper and lower bounds for the 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.

How to cite

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Ern, Alexandre, and Guermond, Jean-Luc. "Evaluation of the condition number in linear systems arising in finite element approximations." ESAIM: Mathematical Modelling and Numerical Analysis 40.1 (2006): 29-48. <http://eudml.org/doc/249710>.

@article{Ern2006,
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. },
author = {Ern, Alexandre, Guermond, Jean-Luc},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite elements; condition number; partial differential equations; linear algebra.; finite elements},
language = {eng},
month = {2},
number = {1},
pages = {29-48},
publisher = {EDP Sciences},
title = {Evaluation of the condition number in linear systems arising in finite element approximations},
url = {http://eudml.org/doc/249710},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Ern, Alexandre
AU - Guermond, Jean-Luc
TI - Evaluation of the condition number in linear systems arising in finite element approximations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/2//
PB - EDP Sciences
VL - 40
IS - 1
SP - 29
EP - 48
AB - 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.
LA - eng
KW - Finite elements; condition number; partial differential equations; linear algebra.; finite elements
UR - http://eudml.org/doc/249710
ER -

References

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  13. J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles de type elliptique, voisine de la variationnelle. Ann. Scuola Norm. Sup. Pisa16 (1962) 305–326.  Zbl0112.33101
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