Local Relaxation Fast Poisson Methods on Hierarchical Meshes

TL;DR

This paper introduces a hierarchical local relaxation method that accelerates the convergence of solving Poisson's equations on hierarchical meshes, achieving high efficiency and accuracy without solving linear systems.

Contribution

The novel hierarchical local relaxation method improves convergence speed for curl-free iterations in Poisson's equations using a mesh hierarchy, avoiding linear system solutions.

Findings

Achieves $\\mathcal{O}(N\log N)$ complexity

Demonstrates high accuracy and efficiency in numerical tests

Effective in solving Poisson--Boltzmann and Poisson--Nernst--Planck equations

Abstract

The local relaxation algorithm is promising for fast solution of Poisson's equations, which computes the electric field distribution in a stepwise manner via local curl-free updates while strictly enforcing Gauss's law. We propose a novel hierarchical local relaxation (HLR) method for speeding up the convergence of curl-free iterations. The local algorithm reformulates the Poisson's equation into the electric-field form and sweeps each cell to minimize the associate electric energy, avoiding the solution of linear systems. The updates with hierarchical meshes significantly accelerate the slow convergence of low-frequency components of the residual in the local curl-free update process. Convergence analysis is performed to obtain the convergence of the hierarchical relaxation approaches. Numerical results show that the HLR methods have the nice properties in accuracy and efficiency and the hierarchical construction leads to an overall computational complexity of $\mathcal{O}(N\log N)$ with respect to the number of grid points. Particularly, the applications in solving the Poisson--Boltzmann and Poisson--Nernst--Planck equations demonstrate the attractive performance for problems which frequently solve the Poisson's equations.