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

Authors: Li Ming; Li Chen-Liang
Journal: Journal of Applied Mathematics
 (2014)

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## 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.


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## Body

## 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 [3], electromagnetics [4], semiconductor modeling [5], and electrical power network modeling [6].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,on⁡∂Ω,
where Ω∈R2 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 [7] and then analyzed by Shi et al. (see [8–11]) and Shaidurov (see [12]). In the recent years, there have been several theoretical analyses and the applications of these methods for the plate bending problems (see [13]), the parabolic problems (see [10]), the nonlinear problems (see [14, 15]), and the Stokes problems (see [16]). 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 [21] 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 [21] 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 Section2. 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 domainΩ=[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 0≤i≤Nx, 0≤j≤Ny. 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 [22]:
(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 solutionsui,j2h and ui,jh on the Z2h grid and the Zh grid are given, respectively (Figure 1). In 2009, Wang and Zhang [21] 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.Figure 1
Four types of points on4×4 grid.The above extrapolation operator is rewritten as the following iterative operatorRET.Algorithm 1.
Consideru~h,new←RET(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,new∈Zh. 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 [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.                      (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 letu¯i, on Zi, i=0,1 denote the exact solutions, the EXCMG method is as following:Algorithm 2 (EXCMG).
Forl=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.


## 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 guessu~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;Step  5. Return to Step  2, if l<L.The difference between RCMG method and EXCMG method is that(17)RCMG=RET+cubicinterpolation+classicaliterativeoperator+CMG,EXCMG=newextrapolation+quadraticinterpolation+classicaliterativeoperator+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 solutionu=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 solutionu=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)cos⁡2(x)-2πsin(πy)cos⁡(πx2)(cos⁡(sin(x))-1)sin(πx2)+1+(4π2x2sin(πy)cos⁡2(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 operatorS 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 grids16×16 and 32×32 are given. We compare EXCMG method with RCMG method for the initial error ∥Err640∥=∥u640-u64∥ on grid 64×64.From Figure2, 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.Example4, grid 64×64, initial error of EXCMG ((a) scale 10-4) and RCMG ((b) scale 10-5).
(a)
(b)


### 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 Figures3 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.Table 1
Numerical results of EXCMG and RCMG for Example4.
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
1.15Table 2
Numerical results of EXCMG and RCMG for Example5.
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
1.06Figure 3
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.Figure 4
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.


## 5.1.  Comparison of the Initial Errors

Assume that the exact solutions of the difference equation on grids16×16 and 32×32 are given. We compare EXCMG method with RCMG method for the initial error ∥Err640∥=∥u640-u64∥ on grid 64×64.From Figure2, 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.Example4, grid 64×64, initial error of EXCMG ((a) scale 10-4) and RCMG ((b) scale 10-5).
(a)
(b)


## 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 Figures3 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.Table 1
Numerical results of EXCMG and RCMG for Example4.
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
1.15Table 2
Numerical results of EXCMG and RCMG for Example5.
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
1.06Figure 3
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.Figure 4
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 [21]. The numerical results show that RCMG method has higher computational accuracy and higher efficiency.



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*Source: 490540-2014-03-10.xml*