Richardson Cascadic Multigrid Method for 2D Poisson Equation Based on a Fourth Order Compact Scheme

Li Ming,Li Chen-Liang

doi:10.1155/2014/490540

Abstract

Based on a fourth order compact difference scheme, a Richardson cascadic multigrid (RCMG) method for 2D Poisson equation is proposed, in which the an initial value on the each grid level is given by the Richardson extrapolation technique (Wang and Zhang (2009)) and a cubic interpolation operator. The numerical experiments show that the new method is of higher accuracy and less computation time.

Highlights

Poisson equation is a partial differential equation (PDE) with broad applications in theoretical physics, mechanical engineering and other fields, such as groundwater flow [1, 2], fluid pressure prediction [3], electromagnetics [4], semiconductor modeling [5], and electrical power network modeling [6].We consider the following two-dimensional (2D) Poisson equation:∂2u (x, y) ∂2u (x, y)− ∂x2 − ∂y2 = f (x, y), in Ω, (1)u (x, y) = 0, on ∂Ω, where Ω ∈ R2 is a rectangular domain or union of rectangular domains with Dirichlet boundary ∂Ω
In order to improve the efficiency of the CMG, some new extrapolation formulas and extrapolation cascadic multigrid (EXCMG) methods are proposed by Chen et al
Based on an asymptotic expansion of finite element method, a new extrapolation formula and an extrapolation cascadic multigrid (EXCMG) method are proposed by Chen et al

Summary

Introduction

Poisson equation is a partial differential equation (PDE) with broad applications in theoretical physics, mechanical engineering and other fields, such as groundwater flow [1, 2], fluid pressure prediction [3], electromagnetics [4], semiconductor modeling [5], and electrical power network modeling [6]. In order to improve the efficiency of the CMG, some new extrapolation formulas and extrapolation cascadic multigrid (EXCMG) methods are proposed by Chen et al (see [17,18,19,20]). These new methods can provide a better initial value for smoothing operator on the refined grid level to accelerate their convergence rate. In order to develop a more efficient CMG method, we use the Richardson extrapolation technique presented in [21] and a new extrapolation formula; a new Richardson extrapolation cascadic multigrid (RCMG) method for 2D Poisson equation is proposed.

Fourth Order Compact Difference Scheme and Richardson Extrapolation Technique

New Extrapolation Formula and EXCMG Method

Richardson Cascadic Multigrid Method

Numerical Experiment and Comparison

Conclusion