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 the superconvergence property and suggest that it also holds for the lowest order Brezzi-Douglas-Marini approximation.
Mots clés : eigenvalue problem, mixed finite element, superconvergence result
@article{M2AN_2009__43_5_853_0, author = {Gardini, Francesca}, title = {Mixed approximation of eigenvalue problems : a superconvergence result}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {853--865}, publisher = {EDP-Sciences}, volume = {43}, number = {5}, year = {2009}, doi = {10.1051/m2an/2009005}, mrnumber = {2559736}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2009005/} }
TY - JOUR AU - Gardini, Francesca TI - Mixed approximation of eigenvalue problems : a superconvergence result JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 853 EP - 865 VL - 43 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009005/ DO - 10.1051/m2an/2009005 LA - en ID - M2AN_2009__43_5_853_0 ER -
%0 Journal Article %A Gardini, Francesca %T Mixed approximation of eigenvalue problems : a superconvergence result %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 853-865 %V 43 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009005/ %R 10.1051/m2an/2009005 %G en %F M2AN_2009__43_5_853_0
Gardini, Francesca. Mixed approximation of eigenvalue problems : a superconvergence result. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 853-865. doi : 10.1051/m2an/2009005. http://www.numdam.org/articles/10.1051/m2an/2009005/
[1] Sobolev spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975). | MR | Zbl
,[2] A posteriori error estimates in finite element acoustic analysis. J. Comput. Appl. Math. 117 (2000) 105-119. | MR | Zbl
, and ,[3] Accurate pressure post-process of a finite element method for elastoacoustics. Numer. Math. 98 (2004) 389-425. | MR | Zbl
, , and ,[4] Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7-32. | Numdam | MR | Zbl
and ,[5] Eigenvalue Problems, in Handbook of Numerical Analysis 2, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991). | MR | Zbl
and ,[6] On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 25 (1997) 131-154. | Numdam | MR | Zbl
, and ,[7] Edge element computation of Maxwell's eigenvalues on general quadrilateral meshes. Math. Models Methods Appl. Sci. 16 (2006) 265-273. | MR | Zbl
, and ,[8] Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68 (1994) 311-324. | MR | Zbl
,[9] Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). | MR | Zbl
and ,[10] The finite element method for elliptic problems, Studies in Mathematics and its application 4. North Holland, Amsterdam (1978). | MR | Zbl
,[11] A posteriori error estimations for mixed approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 9 (1999) 1165-1178. | MR | Zbl
, and ,[12] A posteriori error estimates for eigenvalue problems in mixed form. Ist. lombardo Accd. Sci. Lett. Rend. A. 138 (2004) 17-34.
,[13] A posteriori error estimates for an eigenvalue problem arising from fluid-structure interactions, Computational Fluid and Solid Mechanics. Elsevier, Amsterdam (2005).
,[14] A posteriori error estimates for eigenvalue problems in mixed form. Ph.D. Thesis, Università degli Studi di Pavia, Pavia, Italy (2005).
,[15] Elliptic problem in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Boston (1985). | MR | Zbl
,[16] Zbl
and , Problèmes aux limites non homogènes et applications, Travaux et Recherches Matheḿatiques 17. Dunod, Paris (1968). |[17] 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. | MR | Zbl
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