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

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.

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: −𝜕²𝑢𝑥,𝑦𝜕𝑥²−𝜕²𝑢𝑥,𝑦𝜕𝑦²𝑢=𝑓𝑥,𝑦,inΩ,𝑥,𝑦=0,on𝜕Ω, (1) 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 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,𝐿_𝑥]×[0,𝐿_𝑦]. We discretize Ω with uniform mesh sizes ℎ_𝑥=𝐿_𝑥/𝑁_𝑥 and ℎ_𝑦=𝐿_𝑦/𝑁_𝑦 in the 𝑥 and 𝑦 coordinate directions. The mesh points are (𝑥_𝑖,𝑦_𝑗) with 𝑥_𝑖=𝑖ℎ_𝑥 and 𝑦_𝑗=𝑗ℎ_𝑦, and 0≤𝑖≤𝑁_𝑥, 0≤𝑗≤𝑁_𝑦. Let's denote the mesh aspect ratio 𝛾=ℎ_𝑥/ℎ_𝑦, and 𝑢_𝑖,𝑗 be the solution at the grid point (𝑥_𝑖,𝑦_𝑗), we can rewrite the fourth order compact difference scheme of (1) into the following form [22]: 𝑎𝑢_𝑖𝑗+𝑏𝑢_𝑖+1,𝑗+𝑢_{𝑖−1,𝑗}+𝑐𝑢_𝑖,𝑗+1+𝑢_{𝑖,𝑗−1}+𝑑𝑢_{𝑖+1,𝑗+1}+𝑢_{𝑖+1,𝑗−1}+𝑢_{𝑖−1,𝑗+1}+𝑢_{𝑖−1,𝑗−1}=ℎ²_𝑥28𝑓_𝑖,𝑗+𝑓_𝑖+1,𝑗+𝑓_{𝑖−1,𝑗}+𝑓_𝑖,𝑗+𝑓_{𝑖,𝑗−1}. (2) The coefficients in (2) are 𝑎=−101+𝛾²,𝑏=5−𝛾²,𝑐=5𝛾²−1,𝑑=1+𝛾²2. (3) If the domain Ω is subdivided into a sequence of grids 𝑍_𝑙ℎ (or 𝑍_𝑙), 𝑙=0,1,2,…,𝐿 with step length ℎ_𝑙=ℎ/2^𝑙=ℎ_𝑙,𝑥=ℎ_𝑙,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 𝐴^𝑙𝑢^𝑙=𝐹^𝑙,𝑙=0,1,2,…,𝐿. (4)

Assume the fourth order accurate solutions 𝑢^2ℎ_𝑖,𝑗 and 𝑢^ℎ_𝑖,𝑗 on the 𝑍_2ℎ grid and the 𝑍_ℎ grid are given, respectively (Figure 1). In 2009, Wang and Zhang [21] applied the Richardson extrapolation (where 𝑝=4)𝑢^2ℎ_𝑖,𝑗=2^𝑝𝑢^ℎ_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗2^𝑝=−116𝑢^ℎ_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗15 (5) to get a sixth order accurate solution 𝑢^2ℎ_𝑖,𝑗 on 𝑍_2ℎ.

Figure 1 

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,new←RET(𝑢^ℎ,𝑢^2ℎ,𝜀,𝑘^max).
Step  1. Set 𝑢^ℎ,old∶=𝑢^ℎ, 𝑘∶=0.
Step  2. Update every (even, even) grid point on 𝑍_ℎ by Richardson extrapolation formula (see (5)); then use direct interpolation to get 𝑢^ℎ,new_2𝑖,2𝑗∈𝑍_ℎ. Consider 𝑢^ℎ,new_2𝑖,2𝑗∶=16𝑢^ℎ,old_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗.15 (6)Step  3. Update every (odd, odd) grid point on 𝑍_ℎ. From (2), for each (odd, odd) point (𝑖,𝑗), the updated solution is 𝑢^ℎ,new_𝑖,𝑗1∶=𝑎𝐹_𝑖,𝑗−𝑏𝑢^ℎ,old_𝑖+1,𝑗+𝑢^ℎ,old_{𝑖−1,𝑗}−𝑐𝑢^ℎ,old_𝑖,𝑗+1+𝑢^ℎ,old_{𝑖,𝑗−1}−𝑑𝑢^ℎ,new_{𝑖+1,𝑗+1}+𝑢^ℎ,new_{𝑖+1,𝑗−1}+𝑢^ℎ,new_{𝑖−1,𝑗+1}+𝑢^ℎ,new_{𝑖−1,𝑗−1}. (7) Here, 𝐹_𝑖,𝑗 represents the right-hand side part of (2).
Step  4. Update every (odd, even) grid point on 𝑍_ℎ. From (2), for each (odd, even) grid point, the updated value is 𝑢^ℎ,new_𝑖,𝑗1∶=𝑎𝐹_𝑖,𝑗−𝑏𝑢^ℎ,new_𝑖+1,𝑗+𝑢^ℎ,new_{𝑖−1,𝑗}−𝑐𝑢^ℎ,new_𝑖,𝑗+1+𝑢^ℎ,new_{𝑖,𝑗−1}−𝑑𝑢^ℎ,old_{𝑖+1,𝑗+1}+𝑢^ℎ,old_{𝑖+1,𝑗−1}+𝑢^ℎ,old_{𝑖−1,𝑗+1}+𝑢^ℎ,old_{𝑖−1,𝑗−1}. (8)
Step  5. Update every (even, odd) grid point on 𝑍_ℎ. 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||≤𝜀 or 𝑘=𝑘^max, 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. Ex𝑢^ℎ_2𝑖,2𝑗∶=5𝑢^ℎ_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗4,Ex𝑢^ℎ_2𝑖+1,𝑗∶=𝑢^ℎ_2𝑖+1,𝑗+𝑢^ℎ_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗+𝑢^ℎ_{2𝑖+2,2𝑗}−𝑢^2ℎ_𝑖+1,𝑗8,Ex𝑢^ℎ_{2𝑖,2𝑗+1}∶=𝑢^ℎ_{2𝑖,2𝑗+1}+𝑢^ℎ_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗+𝑢^ℎ_{2𝑖,2𝑗+2}−𝑢^2ℎ_𝑖,𝑗+18,Ex𝑢^ℎ_{2𝑖+1,2𝑗+1}∶=𝑢^ℎ_{2𝑖+1,2𝑗+1}+𝑢^ℎ_2𝑖,2𝑗−𝑢^2ℎ_𝑖,𝑗+𝑢^ℎ_{2𝑖+2,2𝑗}−𝑢^2ℎ_𝑖+1,𝑗+𝑢^ℎ_{2𝑖,2𝑗+2}−𝑢^2ℎ_𝑖,𝑗+1+𝑢^ℎ_{2𝑖+2,2𝑗+2}−𝑢^2ℎ_{𝑖+1,𝑗+1}×16⁻¹. (9)

Let us denote the above new extrapolation formula by operator Ex𝑢^ℎ∶=𝐹𝑢^2ℎ,𝑢^ℎ. (10)

Now let 𝑢^𝑖, on 𝑍_𝑖, 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)) Ex𝑢^𝑙−1∶=𝐹𝑢^𝑙−2,𝑢^𝑙−1. (11)
Step  2. Compute the initial value 𝑢^𝑙,0∶=𝐼₂Ex𝑢^𝑙−1 (12) on 𝑍_𝑙 by using quadratic interpolation operator 𝐼₂.
Step  3. Smooth 𝑚_𝑙 times to get the iterative solution 𝑢^𝑙∶=𝑆^𝑚𝑙_𝑙𝑢^𝑙,0 (13) on 𝑍_𝑙 by using some classical iterative operator 𝑆_𝑙.
Step  4. Return to Step  1 if 𝑙<𝐿, until you get the final iterative solution 𝑢^𝐿 on the finest grid 𝑍_𝐿.

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 𝑢^𝑙,0 on the refined grid 𝑍_𝑙ℎ. 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 𝑍_𝑙ℎ. Similar to the standard CMG method, we propose the following Richardson cascadic multigrid (RCMG) method.

Algorithm  3 [RCMG]

Step  1. Exactly solve the equation 𝐴^𝑙𝑢^𝑙=𝐹^𝑙 on coarsest grid 𝑍_𝑙,𝑙=1,2.

Step  2. Run Algorithm 1; we have 𝑢^𝑙=RET𝑢^𝑙,𝑢^𝑙−1,𝜀,𝑘_𝑙^max. (14)

Step  3. Use a cubic interpolation operator 𝐼₃ to have the initial value 𝑢^𝑙+1,0∶=𝐼₃𝑢^𝑙 (15) on the gird level 𝑍_𝑙+1.

Step  4. Smoothing 𝑤_𝑙 times by using the classical iterative operator 𝑆_𝑙, 𝑢^𝑙+1∶=𝑆^𝑤𝑙_𝑙𝑢^𝑙+𝑙,0 (16) on the level 𝑍_𝑙+1. Set 𝑙∶=𝑙+1;

Step  5. Return to Step  2, if 𝑙<𝐿.

The difference between RCMG method and EXCMG method is that RCMG=RET+cubicinterpolation+classicaliterativeoperator+CMG,EXCMG=newextrapolation+quadraticinterpolation+classicaliterativeoperator+CMG. (17)

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−𝑒^𝑥)(1−𝑥²)(1−𝑦²); the forcing function 𝑓=2sin𝑦𝑒^𝑥−1𝑥²−1+2sin𝑦𝑒^𝑥−1𝑦²𝑦𝑒−1−sin^𝑥−1𝑥²−1𝑦²𝑦𝑒−1+4𝑦cos^𝑥−1𝑥²−1+4𝑥𝑒^𝑥sin𝑦𝑦²−1+𝑒^𝑥sin𝑦𝑥²−1𝑦²−1. (18)

Example 5. The exact solution 𝑢=ln(1+sin(𝜋𝑥²))(cos(sin(𝑥))−1)sin(𝜋𝑦); the forcing function 𝑓=𝜋²sin𝜋𝑦logsin𝜋𝑥²(((𝑥)((𝑥+1cossin))−1−sinsin))sin𝜋𝑦logsin𝜋𝑥²(𝑥)((𝑥+1sin+cossin))sin𝜋𝑦logsin𝜋𝑥²+1cos²(𝑥)−2𝜋sin𝜋𝑦cos𝜋𝑥²(cos(sin(𝑥))−1)sin𝜋𝑥²++14𝜋²𝑥²sin𝜋𝑦cos²𝜋𝑥²(cos(sin(𝑥))−1)sin𝜋𝑥²+1²+4𝜋²𝑥²sin𝜋𝑦sin𝜋𝑥²(cos(sin(𝑥))−1)sin𝜋𝑥²+((𝑥(𝑥)+14𝜋𝑥sinsin))sin𝜋𝑦coscos𝜋𝑥²sin𝜋𝑥².+1 (19)
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=8×2^{𝐿−𝑙+1}. And in RCMG, we set the number of iteration 𝑘_𝑙^max (Step  2) and 𝑤_𝑙 (Step  4) be 8×2^𝐿−𝑙. We set 𝜀=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 ‖Err⁰₆₄‖=‖𝑢⁰₆₄−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 [17–20], the RCMG method can obtain good convergence rate.

(a)

(b)

Figure 2 

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.

Table 1 

Numerical results of EXCMG and RCMG for Example 4.

Table 2 

Numerical results of EXCMG and RCMG for Example 5.

Figure 3 

Comparison of the maximum error ‖𝑢𝐿−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 ‖𝑢𝐿−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.

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).