Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility
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Publication:6323635
DOI10.1007/S00211-021-01205-8arXiv1908.05029MaRDI QIDQ6323635
Publication date: 14 August 2019
Abstract: We consider Galerkin approximations of holomorphic Fredholm operator eigenvalue problems for which the operator values don't have the structure "coercive+compact". In this case the regularity (in sense of [O. Karma, Numer. Funct. Anal. Optim. 17 (1996)]) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework immediately improves the results of [T. Hohage, L. Nannen, BIT 55(1) (2015)], is immediately applicable to analyze approximations of eigenvalue problems related to [A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Num. Math. 138(4) (2018)] and is already applied in [G. Unger, preprint (2017)].
Nonlinear spectral theory, nonlinear eigenvalue problems (47J10) Numerical methods for eigenvalue problems for boundary value problems involving PDEs (65N25) Numerical solution of nonlinear eigenvalue and eigenvector problems (65H17)
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