On the adaptive finite element analysis of the Kohn-Sham equations: methods, algorithms, and implementation

• Published in: International journal for numerical methods in engineering Vol. 106; no. 11; pp. 863 - 888
• Main Authors: Davydov, Denis, Young, Toby D., Steinmann, Paul
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 15.06.2016; Wiley Subscription Services, Inc
• Subjects: adaptive finite element method; Algorithms; atomic and molecular physics; Carbon; Computation; Finite element method; Hydrogen; Kohn-Sham equations; Lithium; Mathematical analysis; Mathematical models
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.5140

Summary In this paper, details of an implementation of a numerical code for computing the Kohn–Sham equations are presented and discussed. A fully self‐consistent method of solving the quantum many‐body problem within the context of density functional theory using a real‐space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co‐aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon‐monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley & Sons, Ltd.