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

TL;DR

This paper investigates the regularity of eigenfunctions in weighted Sobolev spaces for Schrödinger-type problems with singular potentials and demonstrates that an $hp$ discontinuous Galerkin method achieves exponential convergence, confirmed by numerical tests.

Contribution

It establishes the analytic regularity of eigenfunctions for singular potentials and proves exponential convergence of an $hp$ dG method for these problems.

Findings

Eigenfunctions belong to analytic-type weighted Sobolev spaces.

The $hp$ dG method achieves spectral accuracy with exponential convergence.

Numerical tests confirm theoretical convergence rates.

Abstract

We study the regularity in weighted Sobolev spaces of Schr\"{o}dinger-type eigenvalue problems, and we analyse their approximation via a discontinuous Galerkin (dG) $hp$ 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 the an isotropically graded $hp$ 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 the behaviour of the method for varying discretisation parameters.