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Approximation of eigenvalues in mixed form, discrete compactness property, and application to $hp$ mixed finite elements. (English) Zbl 1173.65349
Summary: We discuss the Discrete Compactness Property (DCP) which is a well-known tool for the analysis of finite element approximations of Maxwell’s eigenvalues. We restrict our presentation to Maxwell’s eigenvalues, but the theory applies to more general situations and in particular to mixed finite element schemes that can be written in the framework of de Rham complex and which enjoy suitable compactness properties. We investigate the relationships between DCP and standard mixed conditions for the good approximation of eigenvalues. As a consequence of our theory, the convergence analysis of the rectangular $hp$ version of Raviart-Thomas finite elements for the approximation of Laplace eigenvalues is presented as a corollary of the analogous result for $hp$ edge elements applied to the approximation of Maxwell’s eigenvalues [D. Boffi, M. Costabel, M. Dauge and L. Demkowicz, SIAM J. Numer. Anal. 44, No. 3, 979–1004 (2006; 1122.65110)].
##### MSC:
 65N25 Numerical methods for eigenvalue problems (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)