Avoiding discretization issues for nonlinear eigenvalue problems

arXiv2305.01691MaRDI QIDQ6435091

Alex Townsend, Matthew J. Colbrook

Publication date: 2 May 2023

Abstract: The first step when solving an infinite-dimensional eigenvalue problem is often to discretize it. We show that one must be extremely careful when discretizing nonlinear eigenvalue problems. Using examples, we show that discretization can: (1) introduce spurious eigenvalues, (2) entirely miss spectra, and (3) bring in severe ill-conditioning. While there are many eigensolvers for solving matrix nonlinear eigenvalue problems, we propose a solver for general holomorphic infinite-dimensional nonlinear eigenvalue problems that avoids discretization issues, which we prove is stable and converges. Moreover, we provide an algorithm that computes the problem's pseudospectra with explicit error control, allowing verification of computed spectra. The algorithm and numerical examples are publicly available in $extttinfNEP$, which is a software package written in MATLAB.

Has companion code repository: https://github.com/mcolbrook/infnep

Mathematics Subject Classification ID

Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs (35P30) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Spectrum, resolvent (47A10) Numerical methods for eigenvalue problems for boundary value problems involving PDEs (65N25)

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