Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach

TL;DR

This paper introduces a finite-element method with local polynomial basis functions for solid-state electronic-structure calculations, offering systematic convergence, variable resolution, and suitability for large-scale ab initio computations.

Contribution

It develops a new finite-element approach with local polynomial basis functions for electronic-structure calculations, enabling systematic convergence and efficient parallel computation.

Findings

First fully three-dimensional electronic band structures computed using this method.

Demonstrates the method's ability to produce sparse, structured matrices.

Shows potential for large, accurate ab initio calculations.

Abstract

We present an approach to solid-state electronic-structure calculations based on the finite-element method. In this method, the basis functions are strictly local, piecewise polynomials. Because the basis is composed of polynomials, the method is completely general and its convergence can be controlled systematically. Because the basis functions are strictly local in real space, the method allows for variable resolution in real space; produces sparse, structured matrices, enabling the effective use of iterative solution methods; and is well suited to parallel implementation. The method thus combines the significant advantages of both real-space-grid and basis-oriented approaches and so promises to be particularly well suited for large, accurate ab initio calculations. We develop the theory of our approach in detail, discuss advantages and disadvantages, and report initial results, including the first fully three-dimensional electronic band structures calculated by the method.