Journal of Scientific Computing

, Volume 70, Issue 3, pp 1180–1203

# An Extrapolation Cascadic Multigrid Method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

• Kejia Pan
• Dongdong He
• Hongling Hu
Article

## Abstract

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the $$L^2$$-norm and $$L^{\infty }$$-norm for the solution and its gradient when the exact solution belongs to $$C^6$$. Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

## Keywords

Richardson extrapolation Multigrid method Compact difference scheme Quartic interpolation Poisson equation

65N06 65N55

## Notes

### Acknowledgments

Kejia Pan was supported by the National Natural Science Foundation of China (Nos. 41474103 and 41204082), the National High Technology Research and Development Program of China (No. 2014AA06A602), the Natural Science Foundation of Hunan Province of China (No. 2015JJ3148). Dongdong He was supported by the Natural Science Foundation of China (No. 11402174), the Fundamental Research Funds for the Central Universities, the Program for Young Excellent Talents at Tongji University (No. 2013KJ012) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Hongling Hu was supported by the National Natural Science Foundation of China (No. 11301176). The authors would like to thank the two anonymous reviewers for their helpful comments to improve the paper.

## References

1. 1.
Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. Chapman & Hall, London (1989)
2. 2.
Gupta, M.M.: A fourth-order Poisson solver. J. Comput. Phys. 55, 166–172 (1985)
3. 3.
Gupta, M.M., Kouatchou, J.: Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation. Numer. Methods Part. Differ. Equ. 14, 593–606 (1998)
4. 4.
Spotz, W.F., Carey, G.F.: A high-order compact formulation for the 3D poisson equation. Numer. Methods Part. Differ. Equ. 12, 235–243 (1996)
5. 5.
Sutmann, G., Steffen, B.: High-order compact solvers for the three-dimensional Poisson equation. J. Comput. Appl. Math. 187, 142–170 (2006)
6. 6.
Wang, J., Zhong, W., Zhang, J.: A general meshsize fourth-order compact difference discretization scheme for 3D Poisson equation. Appl. Math. Comput. 183, 804–812 (2006)
7. 7.
Gupta, M.M., Kouatchou, J., Zhang, J.: Comparison of second-order and fourth-order discretization for multigrid Poisson solvers. J. Comput. Phys. 132, 226–232 (1997)
8. 8.
Othman, M., Abdullah, A.R.: An efficient multigrid Poisson solver. Int. J. Comput. Math. 71, 541–553 (1999)
9. 9.
Schaffer, S.: High order multi-grid methods. Math. Comput. 43, 89–115 (1984)
10. 10.
Zhang, J.: Multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization. J. Comput. Phys. 179, 170–179 (2002)
11. 11.
Wang, Y., Zhang, J.: Sixth-order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation. J. Comput. Phys. 228, 137–146 (2009)
12. 12.
Zhang, J.: Fast and high accuracy multigrid solution of the three dimensional Poisson equation. J. Comput. Phys. 143, 449–161 (1998)
13. 13.
Ge, Y.B.: Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation. J. Comput. Phys. 229, 6381–6391 (2010)
14. 14.
McCormick, S.F. (ed.): Multigrid Methods. Frontiers in Applied Mathematics. SIAM, Philadelphia (1987)Google Scholar
15. 15.
Briggs, W.L., McCormick, S.F., Henson, V.E.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000)
16. 16.
Trottenberg, U., Oosterlee, C.W., Schller, A.: Multigrid. Academic Press, London (2001)Google Scholar
17. 17.
Moghaderi, H., Dehghan, M., Hajarian, M.: A fast and efficient two-grid method for solving d-dimensional Poisson equations. Numer. Algorithms 72, 483–537 (2016)
18. 18.
Altas, I., Dym, J., Gupta, M.M., Manohar, R.P.: Multigrid solution of automatically generated high-order discretizations for the biharmonic equation. SIAM J. Sci. Comput. 19, 1575–1585 (1998)
19. 19.
Zhang, J., Sun, H., Zhao, J.J.: High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems. Comput. Methods Appl. Mech. Comput. 191, 4661–4674 (2002)
20. 20.
Ge, Y.B., Cao, F.J.: Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems. J. Comput. Phys. 230, 4051–4070 (2011)
21. 21.
Wang, Y., Zhang, J.: Fast and robust sixth-order multigrid computation for the three-dimensional convection–diffusion equation. J. Comput. Appl. Math. 234, 3496–3506 (2010)
22. 22.
Bornemann, F.A., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numer. Math. 75, 135–152 (1996)
23. 23.
Shaidurov, V.: Some estimates of the rate of convergence for the cascadic conjugate-gradient method. Comput. Math. Appl. 31, 161–171 (1996)
24. 24.
Braess, D., Dahmen, W.: A cascadic multigrid algorithm for the Stokes equations. Numer. Math. 82, 179–191 (1999)
25. 25.
Timmermann, G.: A cascadic multigrid algorithm for semilinear elliptic problems. Numer. Math. 86, 717–731 (2000)
26. 26.
Shaidurov, V., Tobiska, L.: The convergence of the cascadic conjugate- gradient method applied to elliptic problems in domains with re-entrant cor- ners. Math. Comput. 69, 501–520 (2000)
27. 27.
Shaidurov, V., Timmermann, G.: A cascadic multigrid algorithm for semi-linear indefinite elliptic problems. Computing 64, 349–366 (2000)
28. 28.
Shi, Z.C., Xu, X.J.: Cascadic multigrid for parabolic problems. J. Comput. Math. 18, 551–560 (2000)
29. 29.
Braess, D., Deuflhard, P., Lipnikov, K.: A subspace cascadic multigrid method for mortar elements. Computing 69, 205–225 (2002)
30. 30.
Stevenson, R.: Nonconforming finite elements and the cascadic multi-grid method. Numer. Math. 91, 351–387 (2002)
31. 31.
Zhou, S.Z., Hu, H.X.: On the convergence of a cascadic multigrid method for semilinear elliptic problem. Appl. Math. Comput. 159, 407417 (2004)
32. 32.
Du, Q., Ming, P.B.: Cascadic multigrid methods for parabolic problems. Sci. China Ser. A Math. 51, 1415–1439 (2008)
33. 33.
Xu, X.J., Chen, W.B.: Standard and economical cascadic multigrid methods for the mortar finite element methods. Numer. Math. Theory Methods Appl. 2, 180–201 (2009)
34. 34.
Yu, H.X., Zeng, J.P.: A cascadic multigrid method for a kind of semilinear elliptic problem. Numer. Algorithms 58, 143–162 (2011)
35. 35.
Shi, Z.C., Xu, X.J., Huang, Y.Q.: Economical cascadic multigrid method (ECMG). Sci. China Ser. A Math. 50(12), 1765–1780 (2007)
36. 36.
Chen, C.M., Hu, H.L., Xie, Z.Q., et al.: Analysis of extrapolation cascadic multigrid method (EXCMG). Sci. China Ser. A-Math. 51, 1349–1360 (2008)
37. 37.
Chen, C.M., Shi, Z.C., Hu, H.L.: On extrapolation cascadic multigrid method. J. Comput. Math. 29, 684–697 (2011)
38. 38.
Hu, H.L., Chen, C.M., Pan, K.J.: Asymptotic expansions of finite element solutions to Robin problems in $$H^3$$ and their application in extrapolation cascadic multigrid method. Sci. China Math. 57, 687–698 (2014)
39. 39.
Hu, H.L., Chen, C.M., Pan, K.J.: Time-extrapolation algorithm (TEA) for linear parabolic problems. J. Comput. Math. 32, 183–194 (2014)
40. 40.
Pan, K.J., Tang, J.T., Hu, H.L., et al.: Extrapolation cascadic multigrid method for 2.5D direct current resistivity modeling (in Chinese), Chinese. J. Geophys. 55, 2769–2778 (2012)Google Scholar
41. 41.
Pan, K.J., Tang, J.T.: 2.5-D and 3-D DC resistivity modelling using an extrapolation cascadic multigrid method. Geophys. J. Int. 197, 1459–1470 (2014)
42. 42.
Newman, G.A.: A Review of high-performance computational strategies for modeling and imaging of electromagnetic induction data. Surv. Geophys. 35, 85–100 (2014)
43. 43.
Berikelashuili, G., Gupta, M.M., Mirianashvili, M.: Convergence of fourth-order compact difference schemes for three-dimensional convection-diffusion equations. SIAM J. Numer. Anal. 45, 443–455 (2007)
44. 44.
Pan, K.J., He, D.D., Hu, H.L.: A new extrapolation cascadic multigrid method for 3D elliptic boundary value problems on rectangular domains. arXiv preprint arXiv:1506.02983 (2015)
45. 45.
Marchuk, G.I., Shaidurov, V.V.: Difference Methods and Their Extrapolations. Springer, New York (1983)
46. 46.
Neittaanmaki, P., Lin, Q.: Acceleration of the convergence in finite-difference method by predictor corrector and splitting extrapolation methods. J. Comput. Math. 5, 181–190 (1987)
47. 47.
Fößmeier, R.: On Richardson extrapolation for finite difference methods on regular grids. Numer. Math. 55, 451–462 (1989)
48. 48.
Han, G.Q.: Spline finite difference methods and their extrapolation for singular two-point boundary value problems. J. Comput. Math. 11, 289–296 (1993)
49. 49.
Sun, H., Zhang, J.: A high order finite difference discretization strategy based on extrapolation for convection diffusion equations. Numer. Methods Part. Differ. Equ. 20, 18–32 (2004)
50. 50.
Rahul, K., Bhattacharyya, S.N.: One-sided finite-difference approximations suitable for use with Richardson extrapolation. J. Comput. Phys. 219, 13–20 (2006)
51. 51.
Munyakazi, J.B., Patidar, K.C.: On Richardson extrapolation for fitted operator finite difference methods. Appl. Math. Comput. 201, 465–480 (2008)
52. 52.
Tam, C.K.W., Kurbatskii, K.A.: A wavenumber based extrapolation and interpolation method for use in conjunction with high-order finite difference schemes. J. Comput. Phys. 157, 588–617 (2000)
53. 53.
Ma, Y., Ge, Y.: A high order finite difference method with Richardson extrapolation for 3D convection diffusion equation. Appl. Math. Comput. 215, 3408–3417 (2010)
54. 54.
Marchi, C.H., Novak, L.A., Santiago, C.D., et al.: Highly accurate numerical solutions with repeated Richardson extrapolation for 2D Laplace equation. Appl. Math. Model. 37, 7386–7397 (2013)
55. 55.
Collatz, L.: The Numerical Treatment of Differential Equations. Springer, New York (1966)Google Scholar