Abstract
Multigrid waveform relaxation provides fast iterative methods for the solution of time-dependent partial differential equations. In this paper we consider anisotropic problems and extend multigrid methods developed for the stationary elliptic case to waveform relaxation methods for the time-dependent parabolic case. We study line-relaxation, semicoarsening and multiple semicoarsening multilevel methods. A two-grid Fourier–Laplace analysis is used to estimate the convergence of these methods for the rotated anisotropic diffusion equation. We treat both continuous time and discrete time algorithms. The results of the analysis are confirmed by numerical experiments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations (Oxford Univ. Press, Oxford, 1995) chapters 7-9.
G. Horton and S. Vandewalle, A space-time multigrid method for parabolic partial differential equations, SIAM J. Sci. Comput. 16(4) (1995) 848–864.
Z. Jackiewicz, B. Owren and B. Welfert, Pseudospectra of waveform relaxation operators, submitted.
J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The continuous-time case, SIAM J. Numer. Anal. 33(2) (1996) 456–474.
J. Janssen and S. Vandewalle, Multigrid waveform relaxation on spatial finite element meshes: The discrete-time case, SIAM J. Sci. Comput. 17(1) (1996) 135–155.
E. Lelarasmee, A.E. Ruehli, and A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits, IEEE Trans. Comput. Aided Design Integr. Circ. Systems 1 (1982) 131–145.
C. Lubich and A. Ostermann, Multigrid dynamic iteration for parabolic equations, BIT 27 (1987) 216–234.
A. Lumsdaine and D. Wu, Spectra and pseudospectra of waveform relaxation operators, SIAM J. Sci. Comput. 18(1) (1997) 286–304.
U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Statist. Comput. 8 (1997) 459–482.
W.A. Mulder, A new multigrid approach to convection problems, J. Comput. Phys. 83 (1989) 303–323.
C.W. Oosterlee, The convergence of parallel multiblock multigrid methods, Appl. Numer. Math. 19 (1995) 115–128.
C.W. Oosterlee and P. Wesseling, On the robustness of a multiple semi-coarsened grid method, Z. Angew. Math. Mech. 75(4) (1995) 251–257.
S. Ta'asan and H. Zhang, On the multigrid waveform relaxation method, SIAM J. Sci. Comput. 16(5) (1995) 1092–1104.
U. Trottenberg, C.W. Oosterlee and A. Schüller, Multigrid (Academic Press, San Diego, CA, 2001).
S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993).
S. Vandewalle and G. Horton, Fourier mode analysis of the multigrid waveform relaxation and timeparallel multigrid methods, Computing 54 (1995) 317–330.
S. Vandewalle and R. Piessens, Efficient parallel algorithms for solving initial-boundary and timeperiodic parabolic partial differential equations, SIAM J. Sci. Statist. Comput. 13(6) (1992) 1330–1346.
T. Washio and C.W. Oosterlee, Flexible multiple semicoarsening for three-dimensional singularly perturbed problems, SIAM J. Sci. Comput. 19(5) (1998) 1646–1666.
P. Wesseling, An Introduction to Multigrid Methods (Wiley, Chichester, 1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Lent, J., Vandewalle, S. Multigrid Waveform Relaxation for Anisotropic Partial Differential Equations. Numerical Algorithms 31, 361–380 (2002). https://doi.org/10.1023/A:1021191719400
Issue Date:
DOI: https://doi.org/10.1023/A:1021191719400