Eigenvalue problems in a non-Lipschitz domain

GABRIEL ACOSTA, MARÍA G. ARMENTANO

IMA JOURNAL OF NUMERICAL ANALYSIS

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2014 vol. 34 p. 83 - 95

0272-4979

In this paper we analyze piecewise linear finite element approximations of the Laplace eigenvalue problem in the plane domain $Omega = {(x; y) : 0 < x < 1; 0 < y < x^{alpha}}$; which gives for $1 < alpha$  the simplest model of an external cusp. Since $Omega$ is curved and non-Lipschitz, the classical spectral theory can not be applied directly. We present the eigenvalue problem in a proper setting, and relying on known convergence results for the associated source problem with $alpha < 3$, we obtain quasi optimal order of convergence for the eigenpairs.