Eigenvalue problems in a non-Lipschitz domain

Abstract

In this paper we analyse piecewise linear finite element approximations of the Laplace eigenvalue problem in the plane domain Ω = (x,y): 0 < x < 1, 0 < y < x, which gives for 1< the simplest model of an external cusp. Since Ω is curved and non-Lipschitz, the classical spectral theory cannot be applied directly. We present the eigenvalue problem in a proper setting, and relying on known convergence results for the associated source problem with <3, we obtain a quasi-optimal order of convergence for the eigenpairs. © 2013 The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.