  We propose a novel efficient algorithm to solve Poisson equation in irregular
two dimensional domains for electrostatics. It can handle Dirichlet, Neumann or
mixed boundary problems in which the filling media can be homogeneous or
inhomogeneous. The basic idea of the new method is solve the problem in three
steps: (i) First solve the equation $\nabla\cdot\mathbf D=\rho$. The inverse of
the divergence operator in a restricted subspace is found to yield the electric
flux density $\mathbf D$ by a fast direct solver in O(N) operations. The
$\mathbf D$ so obtained is nonunique with indeterminate divergence-free
component. Then the electric field is found by $\mathbf E=\mathbf D/\epsilon$.
But $\nabla\times\mathbf E=0$ for electrostatic field; hence, $\mathbf E$ is
curl free and orthogonal to the divergence free space. (ii) An
orthogonalization process is used to purify the electric field making it curl
free and unique. (iii) Then the potential $\phi$ is obtained by solving $\nabla
\phi=-\mathbf E$ or finding the inverse of the gradient operator in a
restricted subspace by a similar fast direct solver in O(N) operations.
Treatments for both Dirichlet and Neumann boundary conditions are addressed.
Finally, the validation and efficiency are illustrated by several numerical
examples. Through these simulations, it is observed that the computational
complexity of our proposed method almost scales as O(N) where N is the triangle
patch number of meshes. Consequently, this new algorithm is a feasible fast
Poisson solver.
