JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 490540 10.1155/2014/490540 490540 Research Article Richardson Cascadic Multigrid Method for 2D Poisson Equation Based on a Fourth Order Compact Scheme Ming Li 1 Chen-Liang Li 2 Cen Song 1 Department of Mathematics Honghe University Mengzi Yunnan 661100 China uoh.edu.cn 2 School of Mathematics and Computing Science Guilin University of Electronics Technology Guilin, Guangxi 541004 China gliet.edu.cn 2014 1032014 2014 09 09 2013 31 12 2013 02 01 2014 10 3 2014 2014 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.

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.

1. 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 , electromagnetics , semiconductor modeling , and electrical power network modeling .

We consider the following two-dimensional (2D) Poisson equation: (1)-2u(x,y)x2-2u(x,y)y2=f(x,y),inΩ,u(x,y)=0,onΩ, where ΩR2 is a rectangular domain or union of rectangular domains with Dirichlet boundary Ω. The solution u(x,y) and the forcing function f(x,y) 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 ) 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 ). 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 Ω=[0,Lx]×[0,Ly]. We discretize Ω with uniform mesh sizes hx=Lx/Nx and hy=Ly/Ny in the x and y coordinate directions. The mesh points are (xi,yj) with xi=ihx and yj=jhy, and 0iNx, 0jNy. Let's denote the mesh aspect ratio γ=hx/hy, and ui,j be the solution at the grid point (xi,yj), we can rewrite the fourth order compact difference scheme of (1) into the following form : (2)auij+b(ui+1,j+ui-1,j)+c(ui,j+1+ui,j-1)+d(ui+1,j+1+ui+1,j-1+ui-1,j+1+ui-1,j-1)=hx22(8fi,j+fi+1,j+fi-1,j+fi,j+fi,j-1). The coefficients in (2) are (3)a=-10(1+γ2),b=5-γ2,c=5γ2-1,d=(1+γ2)2. If the domain Ω is subdivided into a sequence of grids Zlh (or Zl), l=0,1,2,,L with step length hl=h/2l=hl,x=hl,y (namely, γ=1), by using the fourth order compact difference scheme (see (2)), a series of linear equations of the model problem (1) are given as follows (4)Alul=Fl,l=0,1,2,,L.

Assume the fourth order accurate solutions ui,j2h and ui,jh on the Z2h grid and the Zh grid are given, respectively (Figure 1). In 2009, Wang and Zhang  applied the Richardson extrapolation (where p=4)(5)u~i,j2h=(2pu2i,2jh-ui,j2h)2p-1=(16u2i,2jh-ui,j2h)15 to get a sixth order accurate solution u~i,j2h on Z2h.

Four types of points on 4×4 grid.

The above extrapolation operator is rewritten as the following iterative operator RET.

Algorithm 1.

Consider u~h,newRET(u~h,u~2h,ɛ,kmax).

Step  1. Set u~h,old:=u~h, k:=0.

Step  2. Update every (even, even) grid point on Zh by Richardson extrapolation formula (see (5)); then use direct interpolation to get u~2i,2jh,newZh. Consider (6)u~2i,2jh,new:=(16u2i,2jh,old-ui,j2h)15.Step  3. Update every (odd, odd) grid point on Zh. From (2), for each (odd, odd) point (i,j), the updated solution is (7)u~i,jh,new:=1a[Fi,j-b(u~i+1,jh,old+u~i-1,jh,old)-c(u~i,j+1h,old+u~i,j-1h,old)-d(u~i+1,j+1h,new+u~i+1,j-1h,new+u~i-1,j+1h,new+u~i-1,j-1h,new)]. Here, Fi,j represents the right-hand side part of (2).

Step  4. Update every (odd, even) grid point on Zh. From (2), for each (odd, even) grid point, the updated value is (8)u~i,jh,new:=1a[Fi,j-b(u~i+1,jh,new+u~i-1,jh,new)  -c(u~i,j+1h,new+u~i,j-1h,new)-d(u~i+1,j+1h,old+u~i+1,j-1h,old+u~i-1,j+1h,old+u~i-1,j-1h,old)].

Step  5. Update every (even, odd) grid point on Zh. From (2), the idea is similar to the (odd, even) grid point. Let k:=k+1.

Step  6. If ||u~h,new-u~h,old||  ɛ or k=kmax, stop. Else, let u~h,old:=u~h,new 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 ). The numerical experiments show that the EXCMG method is of high accuracy and efficiency. Now we rewrite the new extrapolation formula as follows. (9)Exu~2i,2jh:=(5u2i,2jh-ui,j2h)4,Exu~2i+1,jh:=u2i+1,jh+[(u2i,2jh-ui,j2h)+(u2i+2,2jh-ui+1,j2h)]8,Exu~2i,2j+1h:=u2i,2j+1h+[(u2i,2jh-ui,j2h)+(u2i,2j+2h-ui,j+12h)]8,Exu~2i+1,2j+1h:=u2i+1,2j+1h+[(u2i,2jh-ui,j2h)+(u2i+2,2jh-ui+1,j2h)+(u2i,2j+2h-ui,j+12h)+(u2i+2,2j+2h-ui+1,j+12h)]×16-1.

Let us denote the above new extrapolation formula by operator (10)Exu~h:=F(u2h,uh).

Now let u¯i, on Zi, i=0,1 denote the exact solutions, the EXCMG method is as following:

Algorithm 2 (EXCMG).

For l=2,,L, consider the following

Step  1. Extrapolate by using the new extrapolation formula (see (10)) (11)Exu~l-1:=F(u¯l-2,u¯l-1).

Step  2. Compute the initial value (12)ul,0:=I2Exu~l-1 on Zl by using quadratic interpolation operator I2.

Step  3. Smooth ml times to get the iterative solution (13)u¯l:=Slmlul,0 on Zl by using some classical iterative operator Sl.

Step  4. Return to Step  1 if l<L, until you get the final iterative solution u¯L on the finest grid ZL.

One of the main tasks in cascadic multigrid method is constructing a suitable interpolation. Based on a new extrapolation-interpolation formula, Chen  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 u~l,0 on the refined grid Zlh. 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 Zlh. Similar to the standard CMG method, we propose the following Richardson cascadic multigrid (RCMG) method.

Algorithm  3 [RCMG]

Step  1. Exactly solve the equation Alul=Fl on coarsest grid Zl,l=1,2.

Step  2. Run Algorithm 1; we have (14)u¯l=RET(ul,ul-1,ɛ,klmax).

Step  3. Use a cubic interpolation operator I3 to have the initial value (15)ul+1,0:=I3u¯l on the gird level Zl+1.

Step  4. Smoothing wl times by using the classical iterative operator Sl, (16)ul+1:=Slwlul+l,0 on the level Zl+1. Set l:=l+1;

The difference between RCMG method and EXCMG method is that (17)RCMG=RET+cubic  interpolation+classical  iterative  operator+CMG,EXCMG=new  extrapolation+quadratic  interpolation+classical  iterative  operator+CMG.

5. Numerical Experiment and Comparison

Numerical experiments are conducted to solve a 2D Poisson equation (1) on the unit square domain [0,1]×[0,1].

Example 4.

The exact solution u=sin(y)(1-ex)(1-x2)(1-y2); the forcing function (18)f=2sin(y)(ex-1)(x2-1)+2sin(y)(ex-1)(y2-1)-sin(y)(ex-1)(x2-1)(y2-1)+4ycos(y)(ex-1)(x2-1)+4xexsin(y)(y2-1)+exsin(y)(x2-1)(y2-1).

Example 5.

The exact solution u=ln(1+sin(πx2))(cos(sin(x))-1)sin(πy); the forcing function (19)f=π2sin(πy)log(sin(πx2)+1)(cos(sin(x))-1)-sin(sin(x))sin(πy)log(sin(πx2)+1)sin(x)+cos(sin(x))sin(πy)log(sin(πx2)+1)cos2(x)-2πsin(πy)cos(πx2)(cos(sin(x))-1)sin(πx2)+1+(4π2x2sin(πy)cos2(πx2)(cos(sin(x))-1))(sin(πx2)+1)2+4π2x2sin(πy)sin(πx2)(cos(sin(x))-1)sin(πx2)+1+4πxsin(sin(x))sin(πy)cos(x)cos(πx2)sin(πx2)+1.

We use the conjugate gradient (CG) method as a smoothing iterative operator S in EXCMG method and RCMG method. In EXCMG method, the number of iterations m^l on each grid level has to increase from finer to coarser grids; in this paper let m^l=8×2L-l+1. And in RCMG, we set the number of iteration klmax (Step  2) and wl (Step  4) be 8×2L-l. We set ɛ=10-8 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 16×16 and 32×32 are given. We compare EXCMG method with RCMG method for the initial error Err640=u640-u64 on grid 64×64.

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 , the RCMG method can obtain good convergence rate.

Example 4, grid 64×64, initial error of EXCMG ((a) scale 10-4) and RCMG ((b) scale 10-5).

5.2. Comparison between EXCMG Method and RCMG Method

Let Error=u¯L-u denote the maximum absolute error between the computed solution u¯L and the exact solution u on the finest grid points. The “cpu” denotes the computing time (unit: second) of EXCMG method and RCMG method.

From Figures 3 and 4 and Tables 1 and 2, we see that, under the same conditions, the RCMG method can obtain higher computational precision and spend less computing time than EXCMG method.

Numerical results of EXCMG and RCMG for Example 4.

L 1 / h L EXCMG RCMG
| | u - L - u | | cpu | | u - L - u | | cpu
3 128 6.28 E - 07 0.37 1.27 E - 07 0.14
256 1.12 E - 07 1.12 1.85 E - 08 0.31
512 1.07 E - 08 4.10 2.58 E - 09 1.12

4 128 4.76 E - 07 0.34 2.61 E - 07 0.08
256 1.27 E - 07 1.19 6.11 E - 08 0.42
512 3.58 E - 08 4.38 6.07 E - 09 1.29

5 128 9.21 E - 07 0.51 3.05 E - 07 0.08
256 1.62 E - 07 1.06 7.82 E - 08 0.31
512 2.72 E - 08 4.15 1.72 E - 08 1.15

Numerical results of EXCMG and RCMG for Example 5.

L 1 / h L EXCMG RCMG
| | u - L - u | | cpu | | u - L - u | | cpu
3 128 2.80 E - 06 0.39 1.77 E - 06 0.16
256 4.50 E - 07 1.22 2.50 E - 07 0.50
512 4.18 E - 08 4.17 3.16 E - 08 1.23

4 128 1.07 E - 05 0.25 4.29 E - 06 0.11
256 1.92 E - 06 1.00 9.28 E - 07 0.30
512 1.71 E - 07 4.23 1.39 E - 07 1.17

5 128 1.74 E - 05 0.28 2.56 E - 06 0.09
256 5.29 E - 06 1.01 2.00 E - 06 0.31
512 7.94 E - 07 4.07 3.35 E - 07 1.06

Comparison of the maximum error u¯L-u and cpu time for Example 4 with L=3, taking step lengths hL=1/m, m=64, 128, 256, and 512, respectively.

Comparison of the maximum error u¯L-u and cpu time for Example 5 with L=3, taking step lengths hL=1/m, m=64, 128, 256, and 512, respectively.

6. Conclusion

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.

Acknowledgments

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).