Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

Martin Halla

doi:10.1007/s00211-021-01205-8

Abstract

We consider Galerkin approximations of eigenvalue problems for holomorphic Fredholm operator functions for which the operators do not have the structure “coercive+compact”. In this case the regularity (in the vocabulary of discrete approximation schemes) 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. The technique is based on the knowledge of a suitable Test function operator. 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 can be successfully applied to analyze e.g. complex scaling/perfectly matched layer methods, problems involving sign-changing coefficients due to meta-materials and also (boundary element) approximations of Maxwell-type equations. We demonstrate the application of our framework to the Maxwell eigenvalue problem for a conductive material.

Highlights

In this article we consider the approximation of eigenvalue problems for holomorphic Fredholm operator functions of the following form: find λ ∈ ⊂ C and nonhomogeneous eigenelements u in a Hilbert space X such that A(λ)u = 0, where it is assumed that A(·) : → L(X ) is an operator function which depends holomor
If the discrete approximation scheme is chosen as a Galerkin scheme, the assumptions of [26,27] reduce to a single non-trivial assumption: the regular approximation property
If the operators are of the form “coercive+compact”, the regularity of Galerkin approximations is unconditionally satisfied [19, (32)]

Summary

Introduction

Ben Dhia, Ciarlet and Carvalho [5,10] for the analysis of finite element methods for equations which involve meta materials Both works [5,24] prove weak T-coercivity and T-compatibility. The original motivation for this article was to provide a framework for the convergence analysis of boundary element discretizations of boundary integral formulations of Maxwell eigenvalue problems. Therein (see [23]) the framework is applied to establish convergence results for radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous open systems These eigenvalue problems are linear, classical theory [3,4] can’t be applied. 3 we report in Theorem 3 convergence results for T (·)-compatible Galerkin approximations of eigenvalue problems for weakly T (·)-.

Weak T-coercivity and T-compatibility

T-Compatible approximation of holomorphic eigenvalue problems

Example of application

Maxwell eigenvalue problem for a conductive material