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Local Relaxation Fast Poisson Methods on Hierarchical Meshes

Zhenli Xu, Qian Yin, Hongyu Zhou

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.

Local Relaxation Fast Poisson Methods on Hierarchical Meshes

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 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.
Paper Structure (14 sections, 6 theorems, 66 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 14 sections, 6 theorems, 66 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Electric-field formulation of the Poisson's equation
  3. Local relaxation method
  4. Discretization
  5. Single mesh relaxation
  6. Hierarchical mesh relaxation
  7. Convergence Analysis
  8. Numerical results
  9. Accuracy and comparison for Poisson's equation with exact solution
  10. Time-dependent Poisson's equation tests
  11. Poisson–Boltzmann equation tests
  12. Poisson-Nernst-Planck equation simulations
  13. Conclusions
  14. Numerical Parameters and Settings

Key Result

Proposition 1

The electrostatic energy functional $\mathcal{F}_{\text{pot}}: L^{2}(\Omega,\mathbb{R}^n)\rightarrow\mathbb{R}$ is convex. Specifically, For any $\bm E_1,\bm E_2\in L^{2}(\Omega,\mathbb{R}^n)$, and $\forall \theta\in(0,1)$ we have The equality holds if and only if $\bm E_1=\bm E_2$.

Figures (7)

  • Figure 1: Schematic of the electric-field update on a single grid cell $(i,j)$ with flux $\eta$.
  • Figure 2: (a) Schematic illustration of relaxation on hierarchical meshes with forward (left), and zigzag (right) HLR. (b) Schematic of the electric-field update on a coarse-grid block covering multiple finest cells.
  • Figure 3: Distribution of the curl residual along the $x$-direction at the cross-section $y = 0.5$ for three relaxation methods. Panels (a)–(c) correspond to single mesh relaxation, forward HLR and zigzag HLR methods, respectively.
  • Figure 4: Average iteration time per time step versus the number of grid points $N$ for three relaxation methods and an FFT-based method, averaged over 100 time steps for homogeneous dielectric permittivity case. The stopping tolerance at each time step is set to: $\varepsilon_{tol}=10^{-7}$ (left), $\varepsilon_{tol}=10^{-9}$ (right).
  • Figure 5: Average iteration time per time step versus the number of grid points $N$ for three relaxation methods averaged over 100 time steps for inhomogeneous dielectric permittivity case. The stopping tolerance at each time step is set to: $\varepsilon_{tol}=10^{-7}$ (left), $\varepsilon_{tol}=10^{-9}$ (right). The dash lines represent the FFT results of Figure \ref{['rhotime']}.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Theorem 2.1
  • Remark 1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Theorem 4.1
  • proof