Real-space multigrid solution of electrostatics problems and the kohn-sham equations

A multigrid method for numerically solving electrostatics and quantum chemical problems in real space is discussed. Multigrid techniques are used to solve both the linear Poisson equation and the nonlinear Kohn-Sham and Poisson-Boltzmann equations. The electrostatic potential, Laplacian, charge densities (electrons and nuclei), Kohn-Sham DFT orbitals, and the self-consistent field potential are all represented discretely on the Cartesian grid. High-order finite differences are utilized to obtain physically reasonable results on modestly sized grids. The method is summarized and numerical results for all-electron atomic and molecular structure are presented. The strengths and weaknesses of the method are discussed with suggested directions for future developments, including a new high-order conservative differencing scheme for accurate composite grid computations which preserves the linear scaling property of the multigrid method.