波瓦松-波茲曼方程之多重網格解法

Abstract

We present a first order accurate method for solving the partial differential equation, Poisson-Boltzmann Equation where the coefficients is assumed to be discontinuous across an interface and the source term is allowed to be a delta function. In one-dimension, we take a finite differential approach. Near the discontinuities, the unknown is approximated by a piecewise function. We extend it in two dimensions by taking a dimension-by-dimension in discretization. The underlying grid is regular. We also present two efficient iterative solvers; the algebraic multigrid method to solve the resulting linear system, and the Newton’s method to solve the corresponding nonlinear equations. The main point of this article is to propose an initialization based on geometric multigrid method to reduce number of Newton’s iterations. We show by numerical experiments that total CPU time is nearly proportional to the number of unknowns.