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.
Poisson equation is a partial differential equation (PDE) with broad applications in theoretical physics, mechanical engineering and other fields, such as groundwater flow [
We consider the following twodimensional (2D) Poisson equation:
Multigrid (MG) method is one of the most effective algorithms to solve the large scale problem. In 1996, cascadic multigrid (CMG) method proposed by Bornemann and Deuflhard [
Based on the Richardson extrapolation technique, Wang and Zhang [
In this paper, in order to develop a more efficient CMG method, we use the Richardson extrapolation technique presented in [
The sections are arranged as follows: the fourth order compact difference scheme and Richardson extrapolation technique are given in Section
For convenience, we consider the rectangular domain
Assume the fourth order accurate solutions
Four types of points on
The above extrapolation operator is rewritten as the following iterative operator
Consider
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. (see [
Let us denote the above new extrapolation formula by operator
Now let
For
One of the main tasks in cascadic multigrid method is constructing a suitable interpolation. Based on a new extrapolationinterpolation formula, Chen [
In this section, we use RET operator and a cubic interpolation to interpolate the initial guess
The difference between RCMG method and EXCMG method is that
Numerical experiments are conducted to solve a 2D Poisson equation (
The exact solution
The exact solution
We use the conjugate gradient (CG) method as a smoothing iterative operator
Assume that the exact solutions of the difference equation on grids
From Figure
Example
Let
From Figures
Numerical results of EXCMG and RCMG for Example


EXCMG  RCMG  


cpu 

cpu  
3 
128 

0.37 

0.14 
256 

1.12 

0.31  
512 

4.10 

1.12  


4 
128 

0.34 

0.08 
256 

1.19 

0.42  
512 

4.38 

1.29  


5 
128 

0.51 

0.08 
256 

1.06 

0.31  
512 

4.15 

1.15 
Numerical results of EXCMG and RCMG for Example


EXCMG  RCMG  


cpu 

cpu  
3 
128 

0.39 

0.16 
256 

1.22 

0.50  
512 

4.17 

1.23  


4 
128 

0.25 

0.11 
256 

1.00 

0.30  
512 

4.23 

1.17  


5 
128 

0.28 

0.09 
256 

1.01 

0.31  
512 

4.07 

1.06 
Comparison of the maximum error
Comparison of the maximum error
In this paper, based on a fourth order compact scheme, we present a Richardson cascadic multigrid method for 2D Poisson problem by using Richardson technique presented by [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (Grant no. 11161014), the National Natural Science Foundation of Yunnan Province (Grant no. 2012FD054), and Scientific Research Starting Foundation for Master or Ph.D. of Honghe University (Grant no. XJ1S0925).