Regular Convergence and Finite Element Methods for Eigenvalue Problems

TL;DR

This paper establishes conditions under which finite element methods reliably approximate eigenvalues of compact operators, focusing on regular convergence of discrete holomorphic operator functions and applying the theory to classical eigenvalue problems.

Contribution

It demonstrates that regular convergence of finite element discretizations of holomorphic Fredholm operator eigenvalue problems follows from approximation properties and compact convergence, enabling eigenvalue convergence analysis.

Findings

Finite element methods achieve eigenvalue convergence under regular convergence conditions.

Regular convergence results apply to Dirichlet and biharmonic eigenvalue problems.

The approach unifies convergence analysis for various finite element schemes.

Abstract

Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose spectral problem can be written as the eigenvalue problem of some holomophic Fredholm operator function. Focusing on the finite element methods (conforming, discontinuous Galerkin, etc.), we show that the regular convergence of discrete holomorphic operator functions follows from the approximation property of the finite element spaces and the compact convergence of the discrete operators in some suitable Sobolev space. The convergence for eigenvalues is then obtained using the discrete approximation theory for the eigenvalue problems of holomorphic Fredholm operator functions. The result can be used to show the convergence of various finite element methods for eigenvalue problems such as the Dirhcilet eigenvalue problem and the biharmonic eigenvalue problem.