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Additive domain decomposition operator splittings – convergence analyses in a dissipative framework

Hansen, Eskil LU and Henningsson, Erik LU (2017) In IMA Journal of Numerical Analysis 37(3). p.1496-1519
Abstract
We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction... (More)
We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions. The theoretical results are finally illustrated by numerical experiments. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
domain decomposition, convergence order, additive splitting schemes, alternating direction implicit schemes, parabolic equations, semilinear evolution equations
in
IMA Journal of Numerical Analysis
volume
37
issue
3
pages
24 pages
publisher
Oxford University Press
external identifiers
  • wos:000405416900016
  • scopus:85019518727
ISSN
1464-3642
DOI
10.1093/imanum/drw043
language
English
LU publication?
yes
id
1a867584-b898-4d44-aa15-c45f9acf9b79
alternative location
http://www.maths.lth.se/na/staff/eskil/dataEskil/articles/Domsplit.pdf
date added to LUP
2016-08-02 16:14:53
date last changed
2018-03-11 04:32:09
@article{1a867584-b898-4d44-aa15-c45f9acf9b79,
  abstract     = {We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions. The theoretical results are finally illustrated by numerical experiments.},
  author       = {Hansen, Eskil and Henningsson, Erik},
  issn         = {1464-3642},
  keyword      = {domain decomposition,convergence order,additive splitting schemes,alternating direction implicit schemes,parabolic equations,semilinear evolution equations},
  language     = {eng},
  number       = {3},
  pages        = {1496--1519},
  publisher    = {Oxford University Press},
  series       = {IMA Journal of Numerical Analysis},
  title        = {Additive domain decomposition operator splittings – convergence analyses in a dissipative framework},
  url          = {http://dx.doi.org/10.1093/imanum/drw043},
  volume       = {37},
  year         = {2017},
}