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Local two-sided bounds for eigenvalues of self-adjoint operators. (English) Zbl 1375.65074

The authors examine the equivalence between two numerical methods for computing local two-sided bounds for eigenvalues of self-adjoint operators. The first, developed by S. Zimmermann and U. Mertins [Z. Anal. Anwend. 14, No. 2, 327–345 (1995; Zbl 0831.35117)], is based on ideas of N. J. Lehmann, H. J. Maehly, and F. Goerische, while the second is a geometrically motivated method of E. B. Davies and M. Plum [IMA J. Numer. Anal. 24, No. 3, 417–438 (2004; Zbl 1062.65056)]. The authors establish a general framework allowing the sharpening of previously known results, they determine explicit convergence estimates, and they use a resonant cavity problem that they studied earlier [SIAM J. Sci. Comput. 36, No. 6, A2887–A2906 (2014; Zbl 1317.78012)] to demonstrate the applicability of the Zimmermann-Mertins method [loc. cit.].

MSC:

65J10 Numerical solutions to equations with linear operators
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
47A10 Spectrum, resolvent
47A50 Equations and inequalities involving linear operators, with vector unknowns
47B25 Linear symmetric and selfadjoint operators (unbounded)
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