Research Article | Open Access
Richardson Cascadic Multigrid Method for 2D Poisson Equation Based on a Fourth Order Compact Scheme
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 [1, 2], fluid pressure prediction , electromagnetics , semiconductor modeling , and electrical power network modeling .
We consider the following two-dimensional (2D) Poisson equation: where is a rectangular domain or union of rectangular domains with Dirichlet boundary . The solution and the forcing function are assumed to be sufficiently smooth.
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  and then analyzed by Shi et al. (see [8–11]) and Shaidurov (see ). In the recent years, there have been several theoretical analyses and the applications of these methods for the plate bending problems (see ), the parabolic problems (see ), the nonlinear problems (see [14, 15]), and the Stokes problems (see ). 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 [17–20]). These new methods can provide a better initial value for smoothing operator on the refined grid level to accelerate their convergence rate.
Based on the Richardson extrapolation technique, Wang and Zhang  presented a multiscale multigrid algorithm. Numerical experiments show that the new method is of higher accuracy solution and higher efficiency.
In this paper, in order to develop a more efficient CMG method, we use the Richardson extrapolation technique presented in  and a new extrapolation formula; a new Richardson extrapolation cascadic multigrid (RCMG) method for 2D Poisson equation is proposed.
The sections are arranged as follows: the fourth order compact difference scheme and Richardson extrapolation technique are given in Section 2. Chen’s new extrapolation formula and EXCMG method are introduced in Section 3. In Section 4, we present the RCMG method. In Section 5, the numerical experiments show the effectiveness of the new method.
2. Fourth Order Compact Difference Scheme and Richardson Extrapolation Technique
For convenience, we consider the rectangular domain . We discretize with uniform mesh sizes and in the and coordinate directions. The mesh points are with and , and , . Let's denote the mesh aspect ratio , and be the solution at the grid point , we can rewrite the fourth order compact difference scheme of (1) into the following form : The coefficients in (2) are If the domain is subdivided into a sequence of grids (or ), with step length (namely, , by using the fourth order compact difference scheme (see (2)), a series of linear equations of the model problem (1) are given as follows
Assume the fourth order accurate solutions and on the grid and the grid are given, respectively (Figure 1). In 2009, Wang and Zhang  applied the Richardson extrapolation (where to get a sixth order accurate solution on .
The above extrapolation operator is rewritten as the following iterative operator .
Algorithm 1. Consider .
Step 1. Set , .
Step 2. Update every (even, even) grid point on by Richardson extrapolation formula (see (5)); then use direct interpolation to get . Consider Step 3. Update every (odd, odd) grid point on . From (2), for each (odd, odd) point , the updated solution is Here, represents the right-hand side part of (2).
Step 4. Update every (odd, even) grid point on . From (2), for each (odd, even) grid point, the updated value is
Step 5. Update every (even, odd) grid point on . From (2), the idea is similar to the (odd, even) grid point. Let .
Step 6. If or , stop. Else, let and return to Step 3.
3. New Extrapolation Formula and EXCMG Method
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 [17–20]). The numerical experiments show that the EXCMG method is of high accuracy and efficiency. Now we rewrite the new extrapolation formula as follows.
Let us denote the above new extrapolation formula by operator
Now let , on , denote the exact solutions, the EXCMG method is as following:
Algorithm 2 (EXCMG). For , consider the following
Step 1. Extrapolate by using the new extrapolation formula (see (10))
Step 2. Compute the initial value on by using quadratic interpolation operator .
Step 3. Smooth times to get the iterative solution on by using some classical iterative operator .
Step 4. Return to Step 1 if , until you get the final iterative solution on the finest grid .
4. Richardson Cascadic Multigrid Method
One of the main tasks in cascadic multigrid method is constructing a suitable interpolation. Based on a new extrapolation-interpolation formula, Chen [17–20] proposed the following extrapolation cascadic multigrid (EXCMG) method, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.
In this section, we use RET operator and a cubic interpolation to interpolate the initial guess on the refined grid . Then a classical iterative operator (such as conjugate gradient method) is used as a smoothing operator to compute the high accuracy solution on the fine grid . Similar to the standard CMG method, we propose the following Richardson cascadic multigrid (RCMG) method.
Algorithm 3 [RCMG]
Step 1. Exactly solve the equation on coarsest grid .
Step 2. Run Algorithm 1; we have
Step 3. Use a cubic interpolation operator to have the initial value on the gird level .
Step 4. Smoothing times by using the classical iterative operator , on the level . Set ;
Step 5. Return to Step 2, if .
The difference between RCMG method and EXCMG method is that
5. Numerical Experiment and Comparison
Numerical experiments are conducted to solve a 2D Poisson equation (1) on the unit square domain .
Example 4. The exact solution ; the forcing function
Example 5. The exact solution ; the forcing function
We use the conjugate gradient (CG) method as a smoothing iterative operator in EXCMG method and RCMG method. In EXCMG method, the number of iterations on each grid level has to increase from finer to coarser grids; in this paper let . And in RCMG, we set the number of iteration (Step 2) and (Step 4) be . We set of RET in the RCMG method (on Step 2).
5.1. Comparison of the Initial Errors
Assume that the exact solutions of the difference equation on grids and are given. We compare EXCMG method with RCMG method for the initial error on grid .
From Figure 2, the accuracy of the initial error on the next grid of RCMG method is higher than EXCMG method. Namely, a better initial value on the fine grid can be got by using RCMG method. Based on the results of the literature [17–20], the RCMG method can obtain good convergence rate.
5.2. Comparison between EXCMG Method and RCMG Method
Let denote the maximum absolute error between the computed solution and the exact solution on the finest grid points. The “” denotes the computing time (unit: second) of EXCMG method and RCMG method.
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 numerical results show that RCMG method has higher computational accuracy and higher efficiency.
Conflict of Interests
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).
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Copyright © 2014 Li Ming and Li Chen-Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.