The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting from semi-discretization of the heat equation in one and two dimensions. This theory and some heuristic arguments predict that preconditioning is most effective for the first few iterations.
@article{bwmeta1.element.bwnjournal-article-zmv26i2p229bwm,
author = {Andrzej Augustynowicz and Zdzis\l aw Jackiewicz},
title = {Frequency analysis of preconditioned waveform relaxation iterations},
journal = {Applicationes Mathematicae},
volume = {26},
year = {1999},
pages = {229-242},
zbl = {0998.65082},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv26i2p229bwm}
}
Augustynowicz, Andrzej; Jackiewicz, Zdzisław. Frequency analysis of preconditioned waveform relaxation iterations. Applicationes Mathematicae, Tome 26 (1999) pp. 229-242. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv26i2p229bwm/
[000] [1] K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Oxford Univ. Press, Oxford, 1995. | Zbl 0838.65073
[001] [2] K. Burrage, Z. Jackiewicz, S. P. Nørsett and R. Renaut, Preconditioning waveform relaxation iterations for differential systems, BIT 36 (1996), 54-76. | Zbl 0864.65049
[002] [3] Z. Jackiewicz, B. Owren and B. D. Welfert, Pseudospectra of waveform relaxation operators, Computers Math. Appl. 36 (1998), 67-85. | Zbl 0932.65080
[003] [4] B. Leimkuhler, Estimating waveform relaxation convergence, SIAM J. Sci. Comput. 14 (1993), 872-889. | Zbl 0787.65039
[004] [5] E. Lelerasmee, A. Ruehli and A. Sangiovanni-Vincentelli, The waveform relaxation method for time domain analysis of large scale integrated circuits, IEEE Trans. on CAD of IC and Systems 1 (1982), 131-145.
[005] [6] U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Statist. Comput. 8 (1987), 459-482. | Zbl 0625.65063
[006] [7] O. Nevanlinna, Remarks on Picard-Lindelöf iteration, Part I, BIT 29 (1989), 328-346. | Zbl 0673.65037