The virtual element method for eigenvalue problems with potential terms on polytopal meshes

TL;DR

This paper extends the virtual element method to eigenvalue problems with potential terms on polytopal meshes, demonstrating its flexibility and optimal convergence for quantum and singular eigenvalue problems.

Contribution

It introduces a VEM approach for eigenvalue problems with potential terms on polytopal meshes, applicable to Schrödinger equations and density functional theory models.

Findings

Provides correct spectral approximation with optimal convergence rates.

Demonstrates flexibility of VEM on complex meshes.

Validates method through numerical experiments on quantum problems.

Abstract

We extend the conforming virtual element method to the numerical resolution of eigenvalue problems with potential terms on a polytopal mesh. An important application is that of the Schrodinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provide a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the Quantum Harmonic Oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.