A Holomorphic Operator Function Approach for the Laplace Eigenvalue Problem Using Discontinuous Galerkin Method

Abstract

The paper presents a holomorphic operator function approach for the Laplace eigenvalue problem using the discontinuous Galerkin method. We rewrite the problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for the discontinuous Galerkin method is proved by using the abstract approximation theory for holomorphic operator functions. We employ the spectral indicator method to compute the eigenvalues. Extensive numerical examples are presented to validate the theory.