Abstract
This chapter concludes the abstract analysis of mixed formulations. After studying the finite dimensional case in Chap. 3 and the infinite-dimensional case in Chap. 4, we analyse here the problem of approximating the infinite dimensional case (that, in practice, will come from a PDE problem) by means of a finite dimensional one, treatable with a computer. We shall see that (apparently) reasonable approximations of a well-posed infinite dimensional problem could produce an ill-posed finite dimensional problem or, more generally, a problem whose solution is not an approximation of the original problem. Hence, the typical results of this chapter will be bounds for the difference between the exact solution (that is, the solution of the original, infinite dimensional problem) and the approximate solution (that is, the solution of the discretised, finite dimensional problem). The approximations considered in this chapter are evidently targeted to the Finite Element spaces introduced in Chap. 2. However, many results could be applied to other types of approximation such as spectral methods or Finite Volume methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.P. Aubin. Approximation of elliptic boundary-value problems. John Wiley and Sons, New York, 1972.
D. Boffi. Finite element approximation of eigenvalue problems. Acta Numer., 19:1–120, 2010.
F. Brezzi. On the existence, uniqueness and approximation of saddle–point problems arising from Lagrangian multipliers. R.A.I.R.O. Anal. Numer., 8:129–151, 1974.
P.G. Ciarlet. Mathematical elasticity. Vol. I. North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity.
P.G. Ciarlet. Mathematical elasticity. Vol. II. North-Holland Publishing Co., Amsterdam, 1997. Theory of plates.
A. El Maliki, M. Fortin, N. Tardieu, and A. Fortin. Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations. Int. J. Numerical Methods in Engineering, 83, 2010.
H.C. Elman, D.J. Silvester, and A.J. Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, 2005.
R.S. Falk and J.E. Osborn. Error estimates for mixed methods. RAIRO Anal. Numér., 14(3):249–277, 1980.
M. Fortin. An analysis of the convergence of mixed finite element methods. R.A.I.R.O. Anal. Numer., 11:341–354, 1977.
M. Fortin. Old and new finite elements for incompressible flows. Int. J. Num. Meth. in Fluids, 1:347–364, 1981.
M. Fortin and R. Glowinski. Augmented Lagrangian methods. North-Holland, Amsterdam, 1983.
M. Fortin and R. Pierre. Stability analysis of discrete generalized Stokes problems. Numer. Methods Partial Differential Equations, 8(4):303–323, 1992.
M. Hestenes. Multiplier and gradient methods. J. Opt. Theory and App., 4:303–320, 1969.
J. Nitsche. Ein Kriterium für die Quasi-Optimalität des Ritzchen Verfahrens. Numer. Math., 11:346–348, 1968.
M.J.D. Powell. A method for non-linear constraints in minimization problems. In R. Fletcher, editor, Optimization. Academic Press, London, 1969.
R. Verfürth. A combined conjugate gradient multi-grid algorithm for the numerical solution of the Stokes problem. IMA J. Numer.Anal., 4:441–455, 1984.
R. Verfürth. Error estimates for a mixed finite element approximation of the Stokes equation. R.A.I.R.O. Anal. Numer., 18:175–182, 1984.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Boffi, D., Brezzi, F., Fortin, M. (2013). Approximation of Saddle Point Problems. In: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36519-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-36519-5_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36518-8
Online ISBN: 978-3-642-36519-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)