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 ISBN 978-3-7001-8258-0 Online Edition Research Article
Bo Gong,
Jiguang Sun
S. 228 - 243 doi:10.1553/etna_vol58s228 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol58s228
Abstract: 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 Published Online: 2023/02/27 13:05:09 Object Identifier: 0xc1aa5576 0x003e1811 Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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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 |