[Paper Review] A novel fast solver for Poisson equation with the Neumann boundary condition

Zu‐Hui Ma, Weng Cho Chew+1 more|arXiv (Cornell University)|Jul 18, 2012

Electromagnetic Scattering and Analysis3 citations

TL;DR

This paper presents a novel fast solver for the 2D Poisson equation with Neumann boundary conditions by decoupling the solution into two steps: first solving for the electric displacement field D using a fast tree solver based on loop-tree decomposition, then recovering the potential φ via another fast solver. The method achieves O(N) complexity and significantly reduces solution time compared to traditional FEM with iterative solvers, demonstrating high efficiency and accuracy in numerical examples.

ABSTRACT

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 $ϕ$. 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 $ϕ$ 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.

Motivation & Objective

• To develop a fast, robust, and efficient numerical method for solving the 2D Poisson equation with Neumann boundary conditions in arbitrary geometries.
• To overcome the limitations of traditional FEM, such as high computational cost and slow convergence, especially for irregular domains.
• To leverage loop-tree decomposition and fast tree solvers to achieve near-linear complexity O(N) for solving Poisson problems in electrostatics.
• To validate the method’s accuracy and efficiency through numerical examples, including analytical benchmarks and complex source distributions.

Proposed method

• The method decomposes the solution into two sequential steps: first solving ∇·D = ρ for the electric displacement field D, then solving ∇φ = -D/ε for the potential φ.
• The displacement field D is expanded in a tree basis, and its coefficients are computed using a fast tree solver with O(N) complexity, leveraging loop-tree decomposition from computational electromagnetics.
• Although the tree basis is not fully curl-free, the method corrects for this by removing divergence-free components through an iterative procedure.
• The potential φ is reconstructed using a second fast tree solver applied to the corrected D field, maintaining overall O(N) complexity.
• The divergence-free field removal step involves solving a symmetric, positive definite system (Gram matrix of loop basis), solved iteratively with CG or GMRES.
• A reference potential is imposed to ensure uniqueness, as required for Neumann problems.

Experimental results

Research questions

• RQ1 Can a fast, direct solver for the 2D Poisson equation with Neumann boundary conditions be developed that scales linearly with problem size?
• RQ2 How can loop-tree decomposition techniques from CEM be adapted to solve the Poisson equation efficiently in irregular domains?
• RQ3 What is the computational performance gain of this method over traditional FEM with iterative solvers for Neumann problems?
• RQ4 To what extent does the method preserve accuracy when the tree basis fails to be fully curl-free?

Key findings

• The proposed method achieves solution times significantly lower than traditional FEM with both CG and GMRES solvers, as demonstrated in complexity comparison plots.
• The computational complexity of the new method approaches O(N), with the dominant cost being the divergence-free field removal step, which depends on iterative solver convergence.
• For the simple Neumann problem with analytical solution, the simulated potential distribution matches the exact solution with high accuracy, as confirmed by line comparisons along y=0.1.
• In the line source problem with complex geometry, the method produced a valid potential distribution consistent with FEM results, validating its robustness in non-rectangular domains.
• GMRES outperformed CG in the iterative phase, requiring fewer iterations for convergence under the same stopping criterion (0.01).
• The method successfully handles the non-uniqueness of Neumann problems by fixing a reference potential at a point, ensuring a unique solution.