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The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators. (English) Zbl 1153.65055

Authors’ summary: This paper discusses the order-preserving convergence for spectral approximation of a self-adjoint completely continuous operator \(T\). Under the condition that the approximate operators \(T_h\) converge to \(T\) in norm, it is proved that the \(k\)-th eigenvalue of \(T_h\) converges to the \(k\)-th eigenvalue of \(T\). (We sort the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements, nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems, and prove that the \(k\)-th approximate eigenvalue obtained by these methods converges to the \(k\)-th exact eigenvalue.

MSC:

65J10 Numerical solutions to equations with linear operators
47A10 Spectrum, resolvent
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
47F05 General theory of partial differential operators
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