Evaluation of the condition number in linear systems arising in finite element approximations
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 29-48.

This paper derives upper and lower bounds for the ℓ 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.

DOI : 10.1051/m2an:2006006
Classification : 65F35, 65N30
Mots-clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {29--48},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     doi = {10.1051/m2an:2006006},
     mrnumber = {2223503},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2006006/}
}
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: Modélisation mathématique et analyse numérique
PY  - 2006
SP  - 29
EP  - 48
VL  - 40
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2006006/
DO  - 10.1051/m2an:2006006
LA  - en
ID  - M2AN_2006__40_1_29_0
ER  - 
%0 Journal Article
%A Ern, Alexandre
%A Guermond, Jean-Luc
%T Evaluation of the condition number in linear systems arising in finite element approximations
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2006
%P 29-48
%V 40
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2006006/
%R 10.1051/m2an:2006006
%G en
%F 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: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 29-48. doi : 10.1051/m2an:2006006. http://www.numdam.org/articles/10.1051/m2an:2006006/

[1] M. Ainsworth, W. Mclean and T. Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling. SIAM J. Numer. Anal. 36 (1999) 1901-1932. | Zbl

[2] I. Babuška and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, MD, 1972). Academic Press, New York (1972) 1-359. | Zbl

[3] R.E. Bank and L.R. Scott, On the conditioning of finite element equations with highly refined meshes. SIAM J. Numer. Anal. 26 (1989) 1383-1384. | Zbl

[4] S.C. Brenner and R.L. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, Texts Appl. Math. 15 (1994). | MR | Zbl

[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978). | MR | Zbl

[6] J.-P. Croisille, Finite volume box schemes and mixed methods. ESAIM: M2AN 34 (2000) 1087-1106. | Numdam | Zbl

[7] J.-P. Croisille and I. Greff, Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations 18 (2002) 355-373. | Zbl

[8] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York. Appl. Math. Ser. 159 (2004) | MR | Zbl

[9] A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. (2005) (in press). | Zbl

[10] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). | MR | Zbl

[11] G.H. Golub and C.F. Van Loan, Matrix Computations. John Hopkins University Press, Baltimore, second edition (1989). | MR | Zbl

[12] C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic equations. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285-312. | Zbl

[13] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles de type elliptique, voisine de la variationnelle. Ann. Scuola Norm. Sup. Pisa 16 (1962) 305-326. | Numdam | Zbl

[14] Y. Saad, Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996). | Zbl

[15] K. Yosida, Functional Analysis, Classics in Mathematics. Springer-Verlag, Berlin (1995). Reprint of the sixth edition (1980). | MR

Cité par Sources :