CaltechAUTHORS
  A Caltech Library Service

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. http://resolver.caltech.edu/CaltechAUTHORS:20170612-134545090

[img] PDF - Published Version
See Usage Policy.

363Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20170612-134545090

Abstract

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/S1064827596305337DOIArticle
http://epubs.siam.org/doi/10.1137/S1064827596305337PublisherArticle
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
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ39
Classification Code:AMS subject classifications: 65M55, 65M12, 65M15, 65Y05
Record Number:CaltechAUTHORS:20170612-134545090
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170612-134545090
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78116
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jun 2017 20:52
Last Modified:22 Aug 2017 00:46

Repository Staff Only: item control page