A discontinuous Galerkin scheme for full-potential electronic structure calculations

TL;DR

An efficient numerical scheme for full-potential electronic structure calculations of periodic systems that has the same philosophy as the widely used (L)APW methods in materials science, but possesses systematically spectral convergence rate.

Abstract

In this paper, we construct an efficient numerical scheme for full-potential electronic structure calculations of periodic systems. In this scheme, the computational domain is decomposed into a set of atomic spheres and an interstitial region, and different basis functions are used in different regions: radial basis functions times spherical harmonics in the atomic spheres and plane waves in the interstitial region. These parts are then patched together by discontinuous Galerkin (DG) method. Our scheme has the same philosophy as the widely used (L)APW methods in materials science, but possesses systematically spectral convergence rate. We provide a rigorous a priori error analysis of the DG approximations for the linear eigenvalue problems, and present some numerical simulations in electronic structure calculations.

Figures (14)

• Figure 2.1: The division of $\Omega$ into atomic spheres $\Omega_{\rm in}$ and a interstitial region $\Omega_{\rm out}$.
• Figure 4.1: (Example 1) Numerical errors of plane waves and DG approximations in the single-atom system.
• Figure 4.2: (Example 1) Eigenfunctions along the $x$-axis obtained by plane waves and DG discretizations.
• Figure 4.3: (Example 1) Numerical errors of DG approximations with respect to $K$.
• Figure 4.4: (Example 1) Numerical errors for different types of radial basis functions.

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• proof
• Remark 3.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.1
• proof