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Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation

Gander, Martin J. and Stuart, Andrew M. (1998) Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation. SIAM Journal on Scientific Computing, 19 (6). pp. 2014-2031. ISSN 1064-8275. doi:10.1137/S1064827596305337.

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Waveform relaxation algorithms for partial differential equations (PDEs) are traditionally obtained by discretizing the PDE in space and then splitting the discrete operator using matrix splittings. For the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach. However, the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh. Motivated by the original development of waveform relaxation in circuit simulation, where the circuits are split in the physical domain into subcircuits, we split the PDE by using overlapping domain decomposition. We prove linear convergence of the algorithm in the continuous case on an infinite time interval, at a rate depending on the size of the overlap. This result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement. The algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings. Our analysis quantifies the empirical observation by Jeltsch and Pohl [SIAM J. Sci. Comput., 16 (1995), pp. 40--49] that the convergence rate of a multisplitting algorithm depends on the overlap. Numerical results are presented which support the convergence theory.

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Additional Information:© 1998 Society for Industrial and Applied Mathematics. Received by the editors June 17, 1996; accepted for publication (in revised form) March 4, 1997; published electronically July 26, 1998. We thank Gene Golub for showing us how to prove Lemma 3.8 and Olavi Nevanlinna, Morten Bjørhus, and Sigitas Keras for many interesting discussions.
Subject Keywords:waveform relaxation, domain decomposition, overlapping Schwarz, multisplitting
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ39
Issue or Number:6
Classification Code:AMS subject classifications: 65M55, 65M12, 65M15, 65Y05
Record Number:CaltechAUTHORS:20170612-134545090
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78116
Deposited By: Tony Diaz
Deposited On:12 Jun 2017 20:52
Last Modified:15 Nov 2021 17:36

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