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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 228-243, 2023/02/27
Regular convergence, together with other types of convergence, have been studied since the 1970s for discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of a compact operator $T$ that can be written as an eigenvalue problem of a holomorphic Fredholm operator function $F(\eta) = T-\frac{1}{\eta} I$. Focusing on finite element methods (conforming, discontinuous Galerkin, non-conforming, etc.), we show that the regular convergence of the discrete holomorphic operator functions $F_n$ to $F$ follows from the compact convergence of the discrete operators $T_n$ to $T$. The convergence of the eigenvalues is then obtained using abstract approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to prove the convergence of various finite element methods for eigenvalue problems such as the Dirichlet eigenvalue problem and the biharmonic eigenvalue problem.
Keywords: regular convergence, finite element methods, eigenvalue problems