Numerical Solutions of Poisson Equation by the CIP-Basis Set Method

An accurate and reliable real space method for the ab initio calculation of electronic-structures of materials has been desired. Historically, the most popular method in this field has been the Plane Wave method. However, because the basis functions of the Plane Wave method are not local in real space, it is inefficient to represent the highly localized inner-shell electron state and it generally give rise to a large dense potential matrix which is difficult to deal with. Moreover, it is not suitable for parallel computers, because it requires Fourier transformations. These limitations of the Plane Wave method have led to the development of various real space methods including finite difference method and finite element method, and studies are still in progress. Recently, we have proposed a new numerical method, the CIP-Basis Set (CIP-BS) method [1], by generalizing the concept of the Constrained Interpolation Profile (CIP) method from the viewpoint of the basis set. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the sub-grid scale solution approaches the local real solution by the constraints from the spatial derivative of the original equation. Thus the solution even on the sub-grid scale becomes consistent with the master equation. By increasing the order of the polynomial, this solution quickly converges. The governing equations are unambiguously discretized into matrix form equations requiring the residuals to be orthogonal to the basis functions via the same procedure as the Galerkin method. We have already demonstrated that the method can be applied to calculations of the band structures for crystals with pseudopotentials. It has been certified that the method gives accurate solutions in the very coarse meshes and the errors converge rapidly when meshes are refined. Although, we have dealt with problems in which potentials are represented analytically, in Kohn-Sham equation the potential is obtained by solving Poisson equation, where the charge density is determined by using wave functions. In this paper, we present the CIP-BS method gives accurate solutions for Poisson equation. Therefore, we believe that the method would be a promising method for solving self-consistent eigenvalue problems in real space.