A novel efficient numerical solution of Poisson equation for arbitrary shapes in two dimensions

TL;DR

The paper introduces a fast, efficient numerical algorithm for solving the Poisson equation in arbitrary 2D shapes, applicable to electrostatics with various boundary conditions, achieving near linear computational complexity.

Contribution

A new algorithm that solves the 2D Poisson equation in irregular domains with O(N) complexity using a three-step process involving direct solvers and orthogonalization.

Findings

The method efficiently handles Dirichlet, Neumann, and mixed boundary conditions.

Numerical examples demonstrate the algorithm's accuracy and computational speed.

The approach scales nearly linearly with mesh size, confirming its practicality.

Abstract

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.