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Convergence of the Dirichlet–Neumann method for semilinear elliptic equations

Engström, Emil LU (2024)
Abstract
The Dirichlet–Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov--Poincaré formulation of the Dirichlet–Neumann method.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Working paper/Preprint
publication status
published
subject
pages
20 pages
publisher
arXiv.org
DOI
10.48550/arXiv.2410.14339
project
Moving domain decomposition methods for parabolic PDEs
Next generation numerical partitioning schemes for time dependent PDEs
language
English
LU publication?
yes
id
099df0b4-b351-4c6f-b01a-cd673ab294f7
date added to LUP
2025-01-31 13:20:00
date last changed
2025-04-04 15:15:41
@misc{099df0b4-b351-4c6f-b01a-cd673ab294f7,
  abstract     = {{The Dirichlet–Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov--Poincaré formulation of the Dirichlet–Neumann method.}},
  author       = {{Engström, Emil}},
  language     = {{eng}},
  month        = {{10}},
  note         = {{Preprint}},
  publisher    = {{arXiv.org}},
  title        = {{Convergence of the Dirichlet–Neumann method for semilinear elliptic equations}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2410.14339}},
  doi          = {{10.48550/arXiv.2410.14339}},
  year         = {{2024}},
}