Nonconforming virtual element method for the Schrödinger eigenvalue problem

Abstract

This study presents an in-depth analysis of the nonconforming virtual element method (VEM) as a novel approach for approximating the eigenvalues of the Schrödinger equation. Central to the strategy is deploying the $L^2$ projection operator to discretize potential terms within the model problem. Through compact operator theory, we rigorously establish the methodology's capability to achieve double-order convergence rates for the eigenvalue spectrum. Addressing the challenge posed by the nonconformity of the discrete space, we redefine the solution operator on a weaker space, which aligns with the Babuška-Osborn compactness framework. A comprehensive set of numerical experiments confirms the theoretical findings, showing the approximation qualities and computational efficiency of the method. A series of potential functions are used to illustrate the various challenges behind the choice of a potential for the simulation of the Schrödinger eigenvalue problem. These results confirm the potential of the nonconforming VEM as a robust and accurate tool for quantum mechanical eigenvalue problems.