# Frequency analysis of preconditioned waveform relaxation iterations

Andrzej Augustynowicz; Zdzisław Jackiewicz

Applicationes Mathematicae (1999)

- Volume: 26, Issue: 2, page 229-242
- ISSN: 1233-7234

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topAugustynowicz, Andrzej, and Jackiewicz, Zdzisław. "Frequency analysis of preconditioned waveform relaxation iterations." Applicationes Mathematicae 26.2 (1999): 229-242. <http://eudml.org/doc/219235>.

@article{Augustynowicz1999,

abstract = {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.},

author = {Augustynowicz, Andrzej, Jackiewicz, Zdzisław},

journal = {Applicationes Mathematicae},

keywords = {error analysis; waveform relaxation; convergence; preconditioning; splitting; Laplace transform; semi-discretization; heat equation},

language = {eng},

number = {2},

pages = {229-242},

title = {Frequency analysis of preconditioned waveform relaxation iterations},

url = {http://eudml.org/doc/219235},

volume = {26},

year = {1999},

}

TY - JOUR

AU - Augustynowicz, Andrzej

AU - Jackiewicz, Zdzisław

TI - Frequency analysis of preconditioned waveform relaxation iterations

JO - Applicationes Mathematicae

PY - 1999

VL - 26

IS - 2

SP - 229

EP - 242

AB - 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.

LA - eng

KW - error analysis; waveform relaxation; convergence; preconditioning; splitting; Laplace transform; semi-discretization; heat equation

UR - http://eudml.org/doc/219235

ER -

## References

top- [1] K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Oxford Univ. Press, Oxford, 1995. Zbl0838.65073
- [2] K. Burrage, Z. Jackiewicz, S. P. Nørsett and R. Renaut, Preconditioning waveform relaxation iterations for differential systems, BIT 36 (1996), 54-76. Zbl0864.65049
- [3] Z. Jackiewicz, B. Owren and B. D. Welfert, Pseudospectra of waveform relaxation operators, Computers Math. Appl. 36 (1998), 67-85. Zbl0932.65080
- [4] B. Leimkuhler, Estimating waveform relaxation convergence, SIAM J. Sci. Comput. 14 (1993), 872-889. Zbl0787.65039
- [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.
- [6] U. Miekkala and O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems, SIAM J. Sci. Statist. Comput. 8 (1987), 459-482. Zbl0625.65063
- [7] O. Nevanlinna, Remarks on Picard-Lindelöf iteration, Part I, BIT 29 (1989), 328-346. Zbl0673.65037

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