Regularity and $hp$ discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials
Provides rigorous convergence guarantees for high-order finite element methods on eigenvalue problems with singular potentials, which are common in quantum mechanics.
The paper proves that eigenfunctions of Schrödinger-type eigenvalue problems with singular potentials belong to analytic-type weighted Sobolev spaces, enabling an $hp$ dG method to achieve exponential convergence. Numerical tests in 2D and 3D confirm the theoretical rates.
We study the regularity in weighted Sobolev spaces of Schrödinger-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.