Abstract
The paper presents a new approximation theory for eigenvalue problems of symmetric sesquilinear forms on continuously embedded subspaces of a Hilbert space. First, approximations of inhomogeneous equations are studied in order to introduce the fundamental approximability and closedness conditions. The approximation of eigenvalue problems, additionally, requires convergent, weakly collectively compact sequences of sesquilinear forms. Under these conditions, the convergence of spectra and resolvent sets, of approximation eigenvalues and eigenspaces is established. Main resultsare the associated discretization error estimates and improved convergence statements for Rayleigh quotients ensuring the quadratic convergence behaviour of eigenvalue approximations. The general theory is applied to finite element methods, difference approximations, penalty methods and singular perturbations.
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Stummel, F. (1977). Approximation Methods for Eigenvalue Problems in Elliptic Differential Equations. In: Bohl, E., Collatz, L., Hadeler, K.P. (eds) Numerik und Anwendungen von Eigenwertaufgaben und Verzweigungsproblemen. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’analyse Numérique, vol 38. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5579-2_7
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DOI: https://doi.org/10.1007/978-3-0348-5579-2_7
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