Matrix‐Free Locally Adaptive Finite Element Solution of Density‐Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning

Abstract

In this paper, we propose a new numerical method to find the ground state energy of a given physical system within the Kohn–Sham density functional theory. The h‐adaptive finite element method is adopted for spatial discretization and implemented with matrix‐free operator evaluation. The ground state energy is found by performing unconstrained minimization with non‐orthogonal orbitals using the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. A geometric multigrid preconditioner is applied to improve the convergence. The clear advantage of the proposed approach is demonstrated on selected examples by comparing the performance to other methods such as preconditioned steepest descent minimization. The proposed method provides a solid framework toward O(N) complexity for the locally adaptive real‐space solution of density functional theory with finite elements.

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

“…6 below. Applications of matrixfree geometric multigrid to continuum mechanics were presented in [18] and to electronic calculations with sparse multivectors in [16,17].…”

Cutoff-Based Modeling of Coulomb Interactions for Atomistic-to-Continuum Multiscale Methods

“…• the Barnes-Hut tree algorithm splits the contributions hierarchically to end up in an O( log ) complexity [6], • the Fast Multipole Method (FMM) reduces this complexity to O( ) by taking into account interactions of multipoles, and • the Particle-Particle Particle-Multigrid (P 3 Mg) applies the multigrid method to the solution of Poisson's equation to calculate the potential energy due to the slowly decaying long-range interactions, resulting in O( ) complexity [41]. A similar approach is also used in quantum mechanical calculations [11].…”

Algorithms and Data Structures for Matrix-Free Finite Element Operators with MPI-Parallel Sparse Multi-Vectors

“…In recent years there has been a growing interest in using nite element (FE) discretizations for problems in quantum mechanics [5, 14, 16-18, 22, 45, 48, 49, 52, 57, 59-62], including an open-source production-ready FE-based implementation [44] of the Density Functional eory (DFT) [30,36]. An FE basis has the following advantages: (i) it is a locally adaptive basis with variational convergence; (ii) it has well-developed rigorous error estimates that can be used to e ciently drive locally adaptive re nement; (iii) a hierarchy of nested functional spaces can be used to formulate geometric multigrid (GMG) preconditioners and solvers [6,17], that are important for the computationally e cient solution of source problems and can also improve convergence of direct minimization methods [17,19]; (iv) an MPI-parallel implementation is achieved by domain decomposition of the mesh and does not require global communication; (v) the FE formulation can be equipped with both periodic and non-periodic boundary conditions; (vi) high order polynomial bases can be e ciently employed using matrix-free sum factorization approaches [39,40]; (vii) pseudo-potentials and all electron calculations can be treated within the same framework. Some of those advantages were employed to develop an open-source DFT-FE code [44].…”