Analytical and numerical assessments of boundary variations in Steklov eigenvalue problems

Abstract

In this study, we aim to analyze the effects of several boundary variations on the spectrum of the simplified and generalized Steklov eigenvalue problems (EVPs) in which the spectral parameter resides on the boundary. We mainly focus on assessing the errors that may occur due to the finite element discretization using elements having straight edges to a curved boundary. In this respect, we analytically and numerically analyze the influence of the change in the boundary such as in uniformly expanded discs or in regular polygons inscribed in the unit disc, on the spectrum of both types of Steklov EVPs. We derive Hadamard type variational formulas for both simple and multiple eigenvalues of the generalized Steklov EVP, and thus provide the convergence of the perturbed-domain solutions to those on the unit disc as boundaries of these domains approach to that of the unit disc. We also provide the finite element analysis of the simplified Steklov EVP together with a proof of convergence based on the spectral theory that is not included in the one which has already been applied to the corresponding generalized case.