Convergence of the adaptive finite element discretization based parallel orbital-updating method for eigenvalue problems

TL;DR

This paper proves the convergence of an adaptive finite element parallel orbital-updating method, which is used for efficiently solving large eigenvalue problems in partial differential operators, especially in electronic structure calculations.

Contribution

It provides the first mathematical convergence analysis for the adaptive finite element parallel orbital-updating method in eigenvalue problems.

Findings

The method converges for clustered eigenvalue problems.

It significantly reduces computational cost.

It enhances parallel scalability.

Abstract

It is significant and challenging to solve eigenvalue problems of partial differential operators when many highly accurate eigenpair approximations are required. The adaptive finite element discretization based parallel orbital-updating method, which can significantly reduce the computational cost and enhance the parallel scalability, has been shown to be efficient in electronic structure calculations. However, there is no any mathematical justification for this method in literature. In this paper, we will show the convergence of the method for clustered eigenvalue problems of linear partial differential operators.