Non-consistent approximations of self-adjoint eigenproblems: Application to the supercell method

TL;DR

This paper develops a theoretical framework for analyzing non-consistent approximations of self-adjoint eigenproblems, demonstrating the supercell method's effectiveness and deriving optimal convergence rates for planewave discretizations.

Contribution

It introduces a general framework for analyzing non-consistent eigenproblem approximations and applies it to prove the supercell method is free of spectral pollution with optimal convergence rates.

Findings

Supercell method is spectral pollution free.

Optimal convergence rates for planewave discretization.

Numerical illustrations support theoretical results.

Abstract

In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution. We then show that the supercell method for perturbed periodic Schr\"odinger operators falls into the scope of our study. We prove that this method is spectral pollution free, and we derive optimal convergence rates for the planewave discretization method, taking numerical integration errors into account. Some numerical illustrations are provided.