A hierarchical splines-based $h$-adaptive isogeometric solver for all-electron Kohn--Sham equation

TL;DR

This paper introduces a high-order hierarchical splines-based adaptive isogeometric solver for the all-electron Kohn--Sham equation, achieving high accuracy with fewer degrees of freedom by adaptively resolving singularities at nuclei.

Contribution

It develops an $h$-adaptive hierarchical spline framework with a residual error indicator for efficient all-electron Kohn--Sham calculations, and demonstrates solver effectiveness and accuracy.

Findings

Achieved $10^{-3}$ Hartree/particle accuracy for methane with 6355 degrees of freedom.

Solver convergence is independent of spline basis order.

Numerical experiments confirm the efficiency of the adaptive framework.

Abstract

In this paper, a novel $h$-adaptive isogeometric solver utilizing high-order hierarchical splines is proposed to solve the all-electron Kohn--Sham equation. In virtue of the smooth nature of Kohn--Sham wavefunctions across the domain, except at the nuclear positions, high-order globally regular basis functions such as B-splines are well suited for achieving high accuracy. To further handle the singularities in the external potential at the nuclear positions, an $h$-adaptive framework based on the hierarchical splines is presented with a specially designed residual-type error indicator, allowing for different resolutions on the domain. The generalized eigenvalue problem raising from the discretized Kohn--Sham equation is effectively solved by the locally optimal block preconditioned conjugate gradient (LOBPCG) method with an elliptic preconditioner, and it is found that the eigensolver's convergence is independent of the spline basis order. A series of numerical experiments confirm the effectiveness of the $h$-adaptive framework, with a notable experiment that the numerical accuracy $10^{-3} \mathrm{~Hartree/particle}$ in the all-electron simulation of a methane molecule is achieved using only $6355$ degrees of freedom, demonstrating the competitiveness of our solver for the all-electron Kohn--Sham equation.