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

top
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 L1(Ω). Numerical simulations are presented to illustrate the theoretical results.

How to cite

top

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

top
  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.  
  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.  
  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.  
  4. S.C. Brenner and R.L. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, Texts Appl. Math.15 (1994).  
  5. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978).  
  6. J.-P. Croisille, Finite volume box schemes and mixed methods. ESAIM: M2AN34 (2000) 1087–1106.  
  7. J.-P. Croisille and I. Greff, Some nonconforming mixed box schemes for elliptic problems. Numer. Methods Partial Differential Equations18 (2002) 355–373.  
  8. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York. Appl. Math. Ser.159 (2004)  
  9. A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. (2005) (in press).  
  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).  
  11. G.H. Golub and C.F. van Loan, Matrix Computations. John Hopkins University Press, Baltimore, second edition (1989).  
  12. C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic equations. Comput. Methods Appl. Mech. Engrg.45 (1984) 285–312.  
  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.  
  14. Y. Saad, Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996).  
  15. K. Yosida, Functional Analysis, Classics in Mathematics. Springer-Verlag, Berlin (1995). Reprint of the sixth edition (1980).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.