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Convergence of approximation methods for eigenvalue problem for two forms. (English) Zbl 0584.65033
The eigenvalue problem \(b(u,v)=\lambda a(u,v)\) \(\forall\) \(v\in V\); \(u\in X\), defined by two bilinear forms on a complex vector space V is considered. The form a is supposed to be symmetric and positive definite, b to be continuous with respect to a; X being the closure of V in the a- norm.
The approximation of the considered problem is defined by a sequence of forms \(a_ n\), \(b_ n\) on the space V, with similar properties as a and b, related to a and b by a number of ordering and convergence conditions. By means of the sequence \(a_ n\), an external approximation \(\{X_ n,r_ n,p_ n\}\) of the space X is defined such that \(X_ n=\bar V\) in the \(a_ n\)-norm. Then the general theory of the spectral approximation via (external) approximation of the space is applied.
The main goal of this paper is Theorem 7 giving the convergence of spectral elements (without error bounds). This result covers the classical method of N. Aronszajn [Proc. Symposium spectral theory differential problems, 179-202 (1955; Zbl 0067.091)] and generalizes the paper of R. D. Brown [Rocky Mt. J. Math. 10, 199-215 (1980; Zbl 0445.49043)] concerning the case when b is symmetric.
Reviewer: K.Moszyński

MSC:
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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References:
[1] N. Aronszajn: Approximation methods for eigenvalues of completely continuous symmetric operator. Proc. of Symposium on Spectral Theory and Differential Equations, Stillwater, Oklahoma, 1951, 179-202.
[2] R. D. Brown: Convergence of approximation methods for eigenvalues of completely continuous quadratic forms. Rocky Mt. J. of Math. 10, No. 1, 1980, 199 - 215. · Zbl 0445.49043 · doi:10.1216/RMJ-1980-10-1-199
[3] N. Dunford J. T. Schwartz: Linear Operators, Spectral Theory. New York, Irterscience 1963. · Zbl 0128.34803
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[6] F. Stummel: Diskrete Konvergenz linear Operatoren, I. Math. Ann. 190, 1970, 45 - 92: II. Math. Z. 120, 1971, 231-264. · Zbl 0203.45301 · doi:10.1007/BF01349967
[7] H. F. Weinberger: Variational methods for eigenvalue approximation. Reg. Conf. Series in appl. math. 15, 1974. · Zbl 0296.49033
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