A Novel Fast Solver for Poisson’s Equation with Neumann Boundary Condition

In this paper, we present a novel fast method to solve Poisson’s equation in an arbitrary two dimensional region with Neumann boundary condition, which are frequently encountered in solving electrostatic boundary problems. The basic idea is to solve the original Poisson’s equation by a two-step procedure. In the first stage, we expand the electric field of interest by a set of tree basis functions and solve it with a fast tree solver in O(N) operations. The field such obtained, however, fails to expand the exact field because the tree basis is not curl-free. Despite of this, we can retrieve the correct electric field by purging the divergence-free field. Next, for the second stage, we find the potential distribution rapidly with a same fast solution of O(N) complexity. As a result, the proposed method dramatically reduces solution time compared with traditional FEM methods. In addition, it is the first time that the loop-tree decomposition technique has been introduced to develop fast Poisson solvers. Numerical examples including electrostatic simulations are presented to demonstrate the efficiency of the proposed method.