A New Multilevel Smoothing Method for Wavelet-Based Algebraic Multigrid Poisson Problem Solver

Fabio Henrique Pereira, Kleber Rogério Moreira Prado, Silvio Ikuyo Nabeta


In contrast to the standard algebraic multigrid, the Wavelet-based Algebraic Multigrid method relies more strongly on the smoothing method because the coarse spaces are chosen a priori. So, it is very important to develop new smoother methods, especially for those cases where the classical Gauss-Seidel smoothing method does not give good results. This paper proposes a new multilevel smoothing approach based on projection technique. The proposed smoothing method was applied to smoothing the error in a linear systems issued from finite element solutions of the elliptic equation and the results compared with those obtained from the Gauss-Seidel method.

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W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, second ed., SIAM, California, 2000.

Q. Chang, Y. S. Wong, and H. Fu, "On the algebraic multigrid method``, J. Comput. Phys. 125:279–292, 1996.

G. Haase, M. Kuhn, and S. Reitzinger, "Parallel Algebraic Multigrid Methods on Distributed Memory Computers``, SIAM J. Sci. Comput., 24:410-427, 2002.

V. E. Henson, and U. M. Yang, "BoomerAMG: a Parallel Algebraic Multigrid Solver and Preconditioner``, Appl. Num. Math., 41(1):155−177, 2002.

T. Mifune, T. Iwashita, and M. Shimasaki, "New algebraic multigrid preconditioning for iterative solvers in electromagnetic finite edge-element analyses,`` IEEE Trans. Magn., 39(3):677–1680, 2003.

B. Engquist and E. Luo, "Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients``, SIAM J. on Numerical Analysis, 34:2254–2273, 1997.

W. L. Wan, "Interface preserving coarsening multigrid for elliptic problems with highly discontinuous coefficients``, Numerical Linear Algebra with Applications, 7(8):727–742, 2000.

Y. Xiao, P. Zhang and S. Shu, "Algebraic multigrid methods for elastic structures with highly discontinuous coefficients``, Mathematics and Computers in Simulation, 76(4):249-262, 2007.

W. L. Briggs and V. E. Henson, "Wavelets and multigrid``, SIAM J. Sci. Comput. 14:506–510, 1993.

B. Engquist and E. Luo, "The multigrid method based on a wavelet transformation and Schur complement``, Unpublished.

A. Rieder, "Multilevel methods based on wavelet decomposition``, East-West J. Numer. Math., 2(4):313-330, 1994.

D. De Leon, Wavelet Operators Applied to Multigrid Methods (Ph.D. Thesis), UCLA Mathematics Department CAM Report 00-22, June 2000.

D. De Leon, "A new wavelet multigrid method``, J. Comput. Appl. Math., 220:674-685, 2008.

D. De Leon, "A Wavelet Multigrid Method Applied to the Stokes and Incompressible Navier-Stokes Problems``, Journal of Mathematical Sciences: Advances and Applications, 1(3):601-622, 2008.

A. Avudainayagam, and C. Vani, "Wavelet based multigrid methods for linear and nonlinear elliptic partial differential equations``, Appl. Math. Comput. 148:307–320, 2004.

F. H. Pereira, S. L. L. Verardi, S. I. Nabeta, "A Wavelet-based Algebraic Multigrid preconditioner for sparse linear systems``, Appl. Math. Comput., 182:1098-1107, 2006.

V. M. Garcıa, L. Acevedo, and A. M. Vidal, "Variants of algebraic wavelet-based multigrid methods: Application to shifted linear systems,`` Appl. Math. Comput., 202(1): 287–299, 2008.

F. H. Pereira, et. al., "A Wavelet-based Algebraic Multigrid Preconditioning for Iterative Solvers in Finite Element Analysis``. IEEE Trans. on Magn., 43(4): 1553-1556, 2007.

F. H. Pereira, M. F. Palin, S. L. L. Verardi, S. I. Nabeta, "A Parallel Wavelet-based Algebraic Multigrid black-box Solver and Preconditioner``, 16th Compumag - Conference on the Computation of Electromagnetic Fields, 2007, Aachen.

F. H. Pereira, S. I. Nabeta, "Wavelet-Based Algebraic Multigrid Method Using the Lifting Technique``, J. of Microwaves, Optoelectronics and Electromagnetic Applications, 9(1): 1-9, 2010.

U. Trottenberg, C. W. Oosterlee, and A. Schuller, Multigrid, Academic Press, New York, 2001, pp. 45-56.

S. Zeng and P. Wesseling, "Numerical Stydy of a Multigrid Method with Four Smoothing Methods for the Incompressible Navier-Stokes Equations in General Coordinates``. In N. D. Melson,Th. A. Manteuffel, and S. F. McCormick, editors, Sixth Copper Mountain Conference on Multigrid Methods, 691-708. NASA Langley Research Center, 1993.

J. Blazek, C.-C. Rossow, N. Kroll and R.C. Swanson, "A comparison of several implicit residual smoothing methods in combination with multigrid``. Lecture Notes in Physics, 414:386-390, 1993.

M. F. Adams, M. Brezina, J. J. Hu, and R. S. Tuminaro, "Parallel multigrid smoothing: Polynomial versus Gauss- Seidel``. J. Comput. Phys., 188:593–610, 2003.

K. Watanabe, S. Fujino, H. Igarashi, "Multigrid Method With Adaptive IDR-Based Jacobi Smoother``, IEEE Trans. on Magn., 47(5):1210-1213, 2011.

F. H. Pereira, A. C. M. Junqueira, R. Perrussel, S. I. Nabeta, "An investigation of the requirements over smoother methods in Wavelet-based Algebraic Multigrid``. In: 13th Biennial IEEE Conference on Electromagnetic Field Computation, Athens, 13:1-4, 2008.

F. H. Pereira, S. I. Nabeta, "A New Multilevel Smoothing Method for the Wavelet-Based Algebraic Multigrid.`` In: 17th Conference on the Computation of Electromagnetic Fields, Florianópolis, 2009.

D. Pusch, Efficient Algebraic Multigrid Preconditioners for Boundary Element Matrices, Phd thesis. Johannes Kepler University Linz, Austria, 1997.

F. H. Pereira, M. M. Afonso, J. A. Vasconcelos, and S. I. Nabeta, "An Efficient Two-Level Preconditioner based on Lifting for FEM-BEM Equations``, J. of Microwaves, Optoelectronics and Electromagnetic Applications, 9(2):78-88, 2010.

A. Jensen, A. la Cour-Harbo, The Discrete Wavelet Transform, Ripples in Mathematics, Springer, Berlin, 2001.

T. K. Sarkar, M. Salazar-Palma, C. W. Michael, Wavelet Applications in Engineering Electromagnetics, Artech House, Boston, 2002.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed, SIAM, Philadelphia, 2003.

O. Axelsson, Iterative solution methods, Cambridge University Press, New York, NY, 1995.

DOI: http://dx.doi.org/10.1590/S2179-10742011000200008


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