Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T16:16:25.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixed approximation of eigenvalue problems:A superconvergence result

Published online by Cambridge University Press:  08 April 2009

Francesca Gardini
Francesca Gardini*
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, 27100 Pavia, Italy. francesca.gardini@unipv.it
Get access

Abstract

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm thesuperconvergence property and suggest that it also holds for the lowest orderBrezzi-Douglas-Marini approximation.

Keywords

Eigenvalue problemmixed finite elementsuperconvergence result
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 853 - 865
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975).
Alonso, A., Dello Russo, A. and Vampa, A., A posteriori error estimates in finite element acoustic analysis. J. Comput. Appl. Math. 117 (2000) 105–119. CrossRef
Alonso, A., Dello Russo, A., Padra, C. and Rodriguez, R., Accurate pressure post-process of a finite element method for elastoacoustics. Numer. Math. 98 (2004) 389–425. CrossRef
Arnold, D.N. and Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7–32. CrossRef
I. Babǔska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis 2, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991).
Boffi, D., Brezzi, F. and Gastaldi, L., On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 25 (1997) 131–154.
Boffi, D., Kikuci, F. and Schöberl, J., Edge element computation of Maxwell's eigenvalues on general quadrilateral meshes. Math. Models Methods Appl. Sci. 16 (2006) 265–273. CrossRef
Brandts, J.H., Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68 (1994) 311–324. CrossRef
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991).
P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its application 4. North Holland, Amsterdam (1978).
Durán, R., Gastaldi, L. and Padra, C., A posteriori error estimations for mixed approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 9 (1999) 1165–1178. CrossRef
Gardini, F., A posteriori error estimates for eigenvalue problems in mixed form. Ist. lombardo Accd. Sci. Lett. Rend. A. 138 (2004) 17–34.
F. Gardini, A posteriori error estimates for an eigenvalue problem arising from fluid-structure interactions, Computational Fluid and Solid Mechanics. Elsevier, Amsterdam (2005).
F. Gardini, A posteriori error estimates for eigenvalue problems in mixed form. Ph.D. Thesis, Università degli Studi di Pavia, Pavia, Italy (2005).
P. Grisvard, Elliptic problem in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Boston (1985).
J-.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Matheḿatiques 17. Dunod, Paris (1968).
Marini, L.D., An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22 (1985) 493–496. CrossRef