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A fast and efficient two-grid method for solving d-dimensional poisson equations

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Abstract

The aim of this paper is to introduce a fast and efficient new two-grid method to solve the d-dimensional (d=1,2,3) Poisson elliptic equations. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The finite difference equations are based on applying a finite difference scheme of two- and four-orders (compact finite difference method) for discretizing the spatial derivative. The obtained linear systems of Poisson elliptic equations have been solved by a new two-grid (NTG) method and we also note that the NTG method which is used for solving the large sparse linear systems is faster and more effective than that of the standard two-grid method. We utilize the local Fourier analysis to show that the spectral radius of the new two-grid method for 1D and 2D models is less than that of the standard two-grid method. As well as, we expand the corresponding algorithm to the new multi-grid method. The numerical examples show the efficiency of the new algorithms for solving the d-dimensional Poisson equations.

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References

  1. Aricò, A., Donatelli, M., Serra-Capizzano, S.: V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl 26, 186–214 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aricò, A., Donatelli, M.: A V-cycle Multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105, 511–547 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson, D.F.: An efficient finite difference method for parameter sensitivities of continuous time markov chains. SIAM J. Numer. Anal. 50, 2237–2258 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ansari, R., Hosseini, K., Darvizeh, A., Daneshian, B.: A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects. Appl. Math. Comput. 219, 4977–4991 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Bakhvalov, N.S.: On the convergence of a relaxation method with natural constraints on the elliptic operator. USSR Comput. Math. Math. Phys. 6, 101–135 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramble, J., Pasciak, J., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comp. 57, 23–45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bramble, J., Pasciak, J., Wang, J., Xu, J.: Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comp. 57, 1–21 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brandt, A.: Multi-level adaptive techniques (MLAT) for fast numerical solution to boundary value problem. In: Proceedings of 3rd Internet. Conf. on Numerical Methods in Fluid Mechanics, (Lecture Notes in Physics 18), Paris, pp. 82–89 (1972)

  9. Brandt, A.: Multilevel adaptive solution to boundary value problems. Math. Comp. 31, 333–390 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brandt, A.: Multigrid techniques: 1984 guide with applications to fluid dynamics, GMD studien Nr. 85, Snakh Augustin (1984)

  11. Brandt, A.: Rigorous local mode analysis of multigrid. In: Preliminary proceedings of the fourth copper mountain conference on multigrid methods, vol. 1, pp. 55–133 (1989)

  12. Brandt, A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for automatic multigrid solution with application in geodetic computations. In: Evans, D. J. (ed.) Sparsity and its Application, pp. 257–284, Cambridge (1984)

  13. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, Society for Industrial and Applied Mathematics, 2nd edn, Philadelphia (2000)

  14. Bolten, M., Donatelli, M., Huckle, T., Kravvaritis, C.: Generalized grid transfer operators for multigrid methods applied on Toeplitz matrices. BIT Numer. Math. 55, 341–366 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Causon, D.M., Mingham, C.G.: Introductory Finite Difference Methods for PDEs. Bookboon (2010)

  16. Cundall, P.A.: Explicit finite difference method in geomechanics. Numerical Methods in Geomechanics, ASCE (2014)

  17. Dai, R., Wang, Y., Zhang, J.: Fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3D convection-diffusion equation. Comput. Math. Appl. 66, 542–559 (2013)

    Article  MathSciNet  Google Scholar 

  18. Dehghan, M., Mohebbi, A.: High-order compact boundary value method for the solution of unsteady convection-diffusion problems. Math. Comput. Simul. 79, 683–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dehghan, M., Mohebbi, A.: Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. Appl. Math. Comput. 180, 575–593 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Robust and optimal multi-iterative techniques for IgA collocation linear systems. Comput. Method. Appl. Mech. Eng. 284, 1120–1146 (2015)

    Article  MathSciNet  Google Scholar 

  21. Donatelli, M., Serra-Capizzano, S., Sesana, D.: Multigrid methods for Toeplitz linear systems with different size reduction. BIT Numer. Math. 52, 305–327 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fedorenko, R.P.: A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys. 1, 1092–1096 (1962)

    Article  MATH  Google Scholar 

  23. Fedorenko, R.P.: The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4, 227–235 (1964)

    Article  MATH  Google Scholar 

  24. Ge, L., Zhang, J.: Accuracy, robustness and efficiency comparison in iterative computation of convection-diffusion equations with boundary layers. Numer. Methods Partial Differential Eq. 16, 379–394 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gupta, M.M., Kouatchou, J., Zhang, J.: A compact multigrid solver for convection-diffusion equations. J. Comput. Phys. 132, 123–129 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gupta, M.M., Kouatchou, J., Zhang, J.: Comparison of second and fourth-order discretizations for multigrid Poisson solver. J. Comput. Phys. 132, 226–232 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gupta, M.M., Manohar, R.P., Stephenson, J.W.: A single cell high-order scheme for the convection-diffusion equation with variable coefficients. Internat. J. Numer. Methods Fluids 4, 641– 651 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gupta, M.M., Manohar, R.P., Stephenson, J.W.: High-order difference schemes for two-dimensional elliptic equations. Numer. Methods Partial Differential Eq. 1, 71–80 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gustafson, B., Kreiss, H., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)

    MATH  Google Scholar 

  30. Hackbusch, W.: Multigrid Methods and Applications. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  31. Hackbusch, W.: Iterative Solution of Large Sparse System of Equations. Springer, New York (1992)

    Google Scholar 

  32. Haupt, L., Stiller, J., Nagel, W.E.: A fast spectral element solver combining static condensation and multigrid techniques. J. Comput. Phys. 255, 384–395 (2013)

    Article  MathSciNet  Google Scholar 

  33. Huang, J.J., Huang, H., Shu, C., Chew, Y.T., Wang, S.L.: Hybrid multiple-relaxation-time lattice-Boltzmann finite difference method for axisymmetric multiphase flows. J. Phys. A: Math. Theor. 46, 055501 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Jouvet, G., Gräser, C.: An adaptive Newton multigrid method for a model of marine ice sheets. J. Comput. Phys. 252, 419–437 (2013)

  35. Karaa, S., Zhang, J.: Analysis of stationary iterative methods for discrete convection-diffusion equation with nine-point compact scheme. J. Comput. Appl. Math. 154, 447–476 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kim, S.: Compact schemes for acoustics in the frequency domain. Math. Comput. Model 37, 1335–1341 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Khelifi, S.C., Méchitoua, N., Hülsemann, F., Magoulès, F.: A hybrid multigrid method for convection-diffusion problems. J. Comput. Appl. Math. 259, 711–719 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kouatchou, J.: Asymptotic stability of a nine-point multigrid algorithm for convection-diffusion equations. Electron Trans. Numer. Anal. 6, 153–161 (1997)

    MathSciNet  MATH  Google Scholar 

  39. Kriventsev, V., Ninokata, H.: An effective, locally exact finite-difference scheme for convection-diffusion problems. Numerical Heat Transfer, Part B: Fundamentals 36, 183–205 (1999)

    Article  Google Scholar 

  40. Li, C.L.: A new parallel cascadic multigrid method. Appl. Math. Comput. 219, 10150–10157 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Li, X., Zhang, L., Wang, S.: A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator. Appl. Math. Comput. 219, 3187–3197 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163–1227 (2014)

    Article  MathSciNet  Google Scholar 

  43. Liszka, T., Orkisz, J.: The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Struct. 11, 83–95 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  44. MacLachlan, S., Oosterlee, C.: Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs. Numer. Linear Algebra Appl. 18, 751–774 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. McCormick, S.F.: Multigrid Methods. Frontires Appl. Math. SIAM (1987)

  46. Mitchell, A.R., Griffiths, D.F.: The finite difference method in partial differential equations. Wiley (1980)

  47. Mitra, A.K.: Finite Difference Method for the Solution of Laplace Equation, preprint, akmitra.public.iastate.edu.

  48. Mohebbi, A., Dehghan, M.: High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Model 51, 537–549 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Morinishi, Y., Koga, K.: Skew-symmetric convection form and secondary conservative finite difference methods for moving grids. J. Comput. Phys. 257, 1081–1112 (2014)

    Article  MathSciNet  Google Scholar 

  50. Nagel, J.R.: Solving the Generalized Poisson Equation Using the Finite-Difference Method (FDM). Department of Electrical and Computer Engineering, University of Utah, Salt Lake City (2011)

    Google Scholar 

  51. Napov, A., Notay, Y.: Smoothing factor, order of prolongation and actual multigrid convergence. Numer. Math. 118, 457–483 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Nickaeen, M., Ouazzi, A., Turek, S.: Newton multigrid least-squares FEM for the VVP formulation of the Navier-Stokes equations. J. Comput. Phys. 256, 416–427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Oliveira, F., Pinto, M.A.V., Marchi, C.H., Araki, L.K.: Optimized partial semicoarsening multigrid algorithm for heat diffusion problems and anisotropic grids. Appl. Math. Model 36, 4665–4676 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Patil, D.V., Premnath, K.N., Banerjee, S.: Multigrid lattice Boltzmann method for accelerated solution of elliptic equations. J. Comput. Phys. 265, 172–194 (2014)

    Article  MathSciNet  Google Scholar 

  55. Perrone, N., Kao, R.: A general finite difference method for arbitrary meshes. Comput. Struct. 5, 45–57 (1975)

    Article  MathSciNet  Google Scholar 

  56. Shah, T.M.: Analysis of the multigrid method. NASA STI/Recon Technical Report N 91 (1989)

  57. Shang, J.S.: High-order compact difference schemes for time-dependent Maxwell equations. J. Comput. Phys. 153, 312–333 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  58. Spotz, W.F.: High-order compact finite difference schemes for computational mechanics. Ph.D. Thesis, University of Texas at Austin, Austin (1995)

    Google Scholar 

  59. Stuben, K.: Algebraic multigrid (AMG): an introduction with applications. GMD-Forschungszentrum Informationstechnik (1999)

  60. Sun, H., Kang, N., Zhang, J., Carlson, E.S.: A fourth-order compact difference scheme on face centered cubic grids with multigrid method for solving 2D convection-diffusion equation. Math. Comput. Simul. 63, 651–661 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  61. Torregrosa, A.J., Hoyas, S., Gil, A., Galache, J.P.G.: A sparse mesh for compact finite difference-Fourier solvers with radius-dependent spectral resolution in circular domains. Comput. Math. Appl. 67, 1309–1318 (2014)

    Article  MathSciNet  Google Scholar 

  62. Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic, New York (2001)

    MATH  Google Scholar 

  63. Tzanos, C.P.: Higher-order difference method with a multigrid approach for the solution of the incompressible flow equations at high Reynolds numbers. Numerical Heat Transfer, Part B 22, 179–198 (1992)

    Article  Google Scholar 

  64. Van der Vegt, J.J.W., Rhebergen, S.: hp-Multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis. J. Comput. Phys. 231, 7537–7563 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  65. Voller, V.R.: Fast implicit finite difference method for the analysis of phase change problems. Numer. Heat Transfer 17, 155–169 (1990)

    Article  Google Scholar 

  66. Wang, T.: Optimal point-wise error estimate of a compact difference scheme for the Klein-Gordon-Schrödinger equation. J. Math. Anal. Appl. 412, 155–167 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  67. Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)

    MATH  Google Scholar 

  68. Wienands, R., Joppich, W.: Practical fourier analysis for multigrid methods. CRC press (2010)

  69. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581–613 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  70. Yuste, S.B., Quintana-Murillo, J.: Fast, Accurate and Robust Adaptive Finite Difference Methods for Fractional Diffusion Equations: The Size of the Time steps does Matter (2014). preprint arXiv: 1403.4434

  71. Zhang, J.: Accelerated high accuracy multigrid solution of the convection-diffusion equation with high Reynolds number. Numer. Methods Partial Differential Eq 13, 77–92 (1997)

    Article  MATH  Google Scholar 

  72. Zhang, J.: On convergence and performance of iterative methods with fourth-order compact schemes. Numer. Methods Partial Differential Eq 14, 262–283 (1998)

    MathSciNet  Google Scholar 

  73. Zhang, J.: Fast and high accuracy multigrid solution of the three-dimensional Poisson equation. J. Comput. Phys. 143, 449–461 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  74. Zhang, J.: An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation. Comm. Numer. Meth. Eng. 14, 209–218 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  75. Zhang, J.: Preconditioned iterative methods and finite difference schemes for convection-diffusion. Appl. Math. Comput. 109, 11–30 (2000)

    MathSciNet  MATH  Google Scholar 

  76. 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)

    Article  MATH  Google Scholar 

  77. 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. Eng. 191, 4661–4674 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  78. Zhang, P.G., Wang, J.P.: A predictor-corrector compact finite difference scheme for Burgers’ equation. Appl. Math. Comput. 219, 892–898 (2012)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Hamid Moghaderi & Mehdi Dehghan

  2. Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran

    Masoud Hajarian

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  1. Hamid Moghaderi
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  2. Mehdi Dehghan
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Correspondence to Mehdi Dehghan.

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Moghaderi, H., Dehghan, M. & Hajarian, M. A fast and efficient two-grid method for solving d-dimensional poisson equations. Numer Algor 72, 483–537 (2016). https://doi.org/10.1007/s11075-015-0057-8

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