  In this paper we present a novel fast method to solve Poisson equation in an
arbitrary two dimensional region with Neumann boundary condition. The basic
idea is to solve the original Poisson problem by a two-step procedure: the
first one finds the electric displacement field $\mathbf{D}$ and the second one
involves the solution of potential $\phi$. The first step exploits loop-tree
decomposition technique that has been applied widely in integral equations
within the computational electromagnetics (CEM) community. We expand the
electric displacement field in terms of a tree basis. Then, coefficients of the
tree basis can be found by the fast tree solver in O(N) operations. Such
obtained solution, however, fails to expand the exact field because the tree
basis is not completely curl free. Despite of this, the accurate field could be
retrieved by carrying out a procedure of divergence free field removal.
Subsequently, potential distribution $\phi$ can be found rapidly at the second
stage with another fast approach of O(N) complexity. As a result, the method
dramatically reduces solution time comparing with traditional FEM with
iterative method. Numerical examples including electrostatic simulations are
presented to demonstrate the efficiency of the proposed method.
