Commun. Comput. Phys., 14 (2013), pp. 276-300. |
Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis: 1-D Lattice Model System Tiao Lu ^{1}, Wei Cai ^{2*}, Jianguo Xin ^{3}, Yinglong Guo ^{4} 1 HEDPS & CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China.2 INS, Shanghai Jiaotong University, Shanghai 200240, P.R. China; and Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA. 3 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA. 4 School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China. Received 29 February 2012; Accepted (in revised version) 24 August 2012 Available online 27 November 2012 doi:10.4208/cicp.290212.240812a Abstract 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 framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body system. Such a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element interfaces. In this paper, we will apply the DG method to the density matrix minimization formulation, a popular approach in the density functional theory of many body Schrodinger equations. The density matrix minimization is to find the minima of the total energy, expressed as a functional of the density matrix \rho(r,r'), approximated by the proposed enriched basis, together with two constraints of idempotency and electric neutrality. The idempotency will be handled with the McWeeny's purification while the neutrality is enforced by imposing the number of electrons with a penalty method. A conjugate gradient method (a Polak-Ribiere variant) is used to solve the minimization problem. Finally, the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems. AMS subject classifications: 35Q40, 65N30, 65Z05, 81Q05Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Density functional theory, density matrix minimization, discontinuous Galerkin method, linear scaling method. *Corresponding author. Email: tlu@math.pku.edu.cn (T. Lu), wcai@uncc.edu (W. Cai), jxin@uncc.edu (J. Xin), longlazio1900@gmail.com (Y. Guo) |