The Steklov eigenvalue problem in a cuspidal domain

Abstract

In this paper we analyze the approximation, by piecewise linear finite elements, of a Steklov eigenvalue problem in a plane domain with an external cusp. This problem is not covered by the literature and its analysis requires a special treatment. Indeed, we develop new trace theorems and we also obtain regularity results for the source counterpart. Moreover, under appropriate assumptions on the meshes, we present interpolation error estimates for functions in fractional Sobolev spaces. These estimates allow us to obtain appropriate convergence results of the source counterpart which, in the context of the theory of compact operator, are a fundamental tool in order to prove the convergence of the eigenpairs. At the end, we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests.