no. 5
Mixed approximation of eigenvalue problems : a superconvergence result
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 853-865

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.

DOI : 10.1051/m2an/2009005
Classification : 65N25, 65N30, 65Q60
Keywords: eigenvalue problem, mixed finite element, superconvergence result 
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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

[1] R.A. Adams, Sobolev spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975). | Zbl | MR

[2] A. Alonso, A. Dello Russo and A. Vampa, A posteriori error estimates in finite element acoustic analysis. J. Comput. Appl. Math. 117 (2000) 105-119. | Zbl | MR

[3] A. Alonso, A. Dello Russo, C. Padra and R. Rodriguez, Accurate pressure post-process of a finite element method for elastoacoustics. Numer. Math. 98 (2004) 389-425. | Zbl | MR

[4] D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19 (1985) 7-32. | Zbl | MR | Numdam

[5] I. Babǔska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis 2, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991). | Zbl | MR

[6] D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 25 (1997) 131-154. | Zbl | MR | Numdam

[7] D. Boffi, F. Kikuci and J. Schöberl, Edge element computation of Maxwell's eigenvalues on general quadrilateral meshes. Math. Models Methods Appl. Sci. 16 (2006) 265-273. | Zbl | MR

[8] J.H. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68 (1994) 311-324. | Zbl | MR

[9] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics 15. Springer-Verlag, New York (1991). | Zbl | MR

[10] P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its application 4. North Holland, Amsterdam (1978). | Zbl | MR

[11] R. Durán, L. Gastaldi and C. Padra, A posteriori error estimations for mixed approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 9 (1999) 1165-1178. | Zbl | MR

[12] F. Gardini, A posteriori error estimates for eigenvalue problems in mixed form. Ist. lombardo Accd. Sci. Lett. Rend. A. 138 (2004) 17-34.

[13] F. Gardini, A posteriori error estimates for an eigenvalue problem arising from fluid-structure interactions, Computational Fluid and Solid Mechanics. Elsevier, Amsterdam (2005).

[14] F. Gardini, A posteriori error estimates for eigenvalue problems in mixed form. Ph.D. Thesis, Università degli Studi di Pavia, Pavia, Italy (2005).

[15] P. Grisvard, Elliptic problem in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Boston (1985). | Zbl | MR

[16] J-.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Matheḿatiques 17. Dunod, Paris (1968). | Zbl

[17] L.D. Marini, 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. | Zbl | MR

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