A new finite element approach for the Dirichlet eigenvalue problem

Abstract

We propose a new finite element approach, which is different than the classic Babuška–Osborn theory, to approximate Dirichlet eigenvalues. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for conforming finite elements is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.