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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.

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 |

47A10 | Spectrum, resolvent |

47A55 | Perturbation theory of linear operators |

47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |

### Keywords:

external approximation; eigenvalue problem; bilinear forms; spectral approximation; convergence### 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 |

[4] | T. Kato: Perturbation Theory for Linear Operators. Sprirger Verlag, Berlin 1966. · Zbl 0148.12601 |

[5] | T. Regińska: External approximation of eigenvalue problems in Banach spaces. RAFRO Numerical Analysis, 1984. · Zbl 0554.65043 |

[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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