Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials

Abstract

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.