Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A stable, direct solver for the gradient equation

Author: Rob Stevenson
Journal: Math. Comp. 72 (2003), 41-53
MSC (2000): Primary 65N30, 65F05, 42C40, 76D05, 35Q30
Published electronically: June 6, 2002
MathSciNet review: 1933813
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new finite element discretization of the equation $\mathbf{grad}\,p =\mathbf{g}$ is introduced. This discretization gives rise to an invertible system that can be solved directly, requiring a number of operations proportional to the number of unknowns. We prove an optimal error estimate, and furthermore show that the method is stable with respect to perturbations of the right-hand side $\mathbf{g}$. We discuss a number of applications related to the Stokes equations.

References [Enhancements On Off] (What's this?)

  • [Bre74] F. Brezzi.
    On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers.
    RAIRO Anal. Numér., 8:129-151, 1974. MR 51:1540
  • [Bre90] S.C. Brenner.
    A nonconforming multigrid method for the stationary Stokes equations.
    Math. Comp., 55(192):411-437, 1990. MR 91d:65167
  • [Bre96] S.C. Brenner.
    Two-level additive Schwarz preconditioners for nonconforming finite element methods.
    Math. Comp., 65(215):897-921, 1996. MR 96j:65117
  • [BY93] F.A. Bornemann and H. Yserentant.
    A basic norm equivalence for the theory of multilevel methods.
    Numer. Math., 64:455-476, 1993. MR 94b:65155
  • [Cro72] M. Crouzeix.
    Résolution numérique des equations de Stokes et Navier-Stokes stationnaires.
    Séminaire d'analyse numérique, Université de Paris VI, 1972.
  • [CSS86] C. Cuvelier, A. Segal, and A. van Steenhoven.
    Finite Element Methods and Navier-Stokes Equations.
    D. Reidel Publishing Company, Dordrecht, 1986. MR 88g:65106
  • [GR79] V. Girault and P.A. Raviart.
    An analysis of a mixed finite element method for the Navier-Stokes equations.
    Numer. Math., 33:235-271, 1979. MR 81a:65100
  • [GR86] V. Girault and P.A. Raviart.
    Finite element methods for Navier-Stokes equations, Theory and Algorithms.
    Springer-Verlag, Berlin, 1986. MR 88b:65129
  • [Ste99] R.P. Stevenson.
    Nonconforming finite elements and the cascadic multi-grid method.
    Technical Report 1120, University of Utrecht, November 1999.
    To appear in Numer. Math., 2002.
  • [Tho81] F. Thomasset.
    Implementation of Finite Element Methods for Navier-Stokes Equations.
    Springer-Verlag, New-York, 1981. MR 84k:76015
  • [Tur94] S. Turek.
    Multigrid techniques for a divergence-free finite element discretization.
    East-West J. Numer. Math., 2(3):229-255, 1994. MR 96c:65195
  • [Urb96] K. Urban.
    Using divergence free wavelets for the numerical solution of the Stokes problem.
    In O. Axelsson and B. Polman, editors, Algebraic Multilevel Iteration Methods with Applications, pages 261-278, University of Nijmegen, 1996. MR 98d:65145

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65F05, 42C40, 76D05, 35Q30

Retrieve articles in all journals with MSC (2000): 65N30, 65F05, 42C40, 76D05, 35Q30

Additional Information

Rob Stevenson
Affiliation: Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands

Keywords: LBB-stability, Stokes equations, multiscale bases, direct solver
Received by editor(s): April 28, 1998
Published electronically: June 6, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society