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Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods. (English) Zbl 1156.65094
The goal of this paper is to obtain some abstract results of spectral approximation that can be applied to a wide class of nonconforming methods for compact or noncompact operators. The consistency results derived by the authors are extensions of the results developed by J. Descloux , N. Nassif and J. Rappaz [RAIRO, Anal. Numér. 12, 97–112 (1978; Zbl 0393.65024); ibid. 12, 113–119 (1978; Zbl 0393.65025)].
The theory presented here allows the analysis of a large class of discontinuous finite element methods when they are used for the approximation of spectral problems. Two representative eigenvalue elliptical problems are discussed in detail: the Steklov eigenvalue problem (in which the eigenvalue parameter appears in the boundary condition) and an eigenvalue problem for a system of partial differential equations. The analysis is carried out for the lowest order Crouzeix-Raviart finite element space.

##### MSC:
 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65J10 Numerical solutions to equations with linear operators 47A10 Spectrum, resolvent 35P15 Estimates of eigenvalues in context of PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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