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A note on the de Rham complex and a discrete compactness property. (English) Zbl 0983.65125
Author’s abstract: The aim of this paper is to review the mathematical analysis of the eigenvalue problem associated with the Maxwell’s system. Our analysis is quite general and can be applied to several families of edge finite element methods. Moreover, we discuss the links between different conditions that guarantee the good approximations of the eigensolutions. In particular, we prove that the commutativity of the de Rham complex implies the discrete compactness introduced by F. Kikuchi.

MSC:
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
58A12 de Rham theory in global analysis
35P15 Estimates of eigenvalues in context of PDEs
78A25 Electromagnetic theory, general
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:
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