Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis: 1-D Lattice Model System (Communications in Computational Physics, 2014)
In the first of a series of papers, we will study a discontinuous Galerkin (DG)framework for many electron quantum systems. The salient feature of this frameworkis the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteedpiecewise polynomial basis in representing the Hamiltonian of the many body system.Such a flexibility is made possible by using the discontinuous Galerkin methodto approximate the Hamiltonian matrix elements with proper constructions of numericalDG fluxes at the finite element interfaces. In this paper, we will apply the DGmethod to the density matrix minimization formulation, a popular approach in thedensity functional theory of many body Schrödinger equations. The density matrixminimization is to find the minima of the total energy, expressed as a functional of thedensity matrix ρ(r,r′), approximated by the proposed enriched basis, together withtwo constraints of idempotency and electric neutrality. The idempotency will be handledwith the McWeeny's purification while the neutrality is enforced by imposing thenumber of electrons with a penalty method. A conjugate gradient method (a Polak-Ribierevariant) is used to solve the minimization problem. Finally, the linear-scalingalgorithm and the advantage of using the local orbital enriched finite element basis inthe DG approximations are verified by studying examples of one dimensional latticemodel systems.