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Discrete compactness for the $$hp$$ version of rectangular edge finite elements. (English) Zbl 1122.65110
Discrete subspaces of $$H^1$$ and $$H(curl)$$ forming a part of the discrete exact sequence are considered. It is in this context that F. Kikuchi [J. Fac. Sci., Univ. Tokyo, Sect. IA 36, No. 3, 479–490 (1989; Zbl 0698.65067)] introduced the fundamental notion of the discrete compactness property which, along with appropriate approximability properties, guarantees the convergence of discrete Maxwell eigenvalues to the exact ones.
The authors analyze the pure method on a single square element and present the general $$hp$$ theory which relies on an $$L^2$$ estimate which is proved thanks to the evaluation of an inf-sup constant on the reference element. The $$hp$$ edge element spaces, which generalize the first family of Nédélec finite elements are used. The hypotheses of the authors allow for a complete $$hp$$ refinement, including the presence of hanging nodes. The pure $$p$$ version of edge elements, being a subset of the authors’ setting, is naturally covered by their analysis. The same proof applies to meshes of quadrilaterals obtained by affine transformation from the reference square (i.e., parallelograms) and more generally to meshes obtained using the so-called algebraic mesh generators.
The authors recall the consequences of the discrete compactness property to the eigenvalue approximation by a Galerkin method: As a result, the $$k$$-th nonzero eigenvalue of the Galerkin discretization converges to the $$k$$-th nonzero Maxwell eigenvalue. The possibility of proving an exponential convergence rate for the discretization of the Maxwell problem by the weighted regularization method is discussed. Finally, the extension of proposed proofs to the situation of general curvilinear polygons, with meshes obtained using algebraic mesh generators is commented.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory 78A25 Electromagnetic theory, general 35Q60 PDEs in connection with optics and electromagnetic theory 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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