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Modified Neumann-Neumann methods for semi- and quasilinear elliptic equations

Engström, Emil LU and Hansen, Eskil LU orcid (2023) p.1-22
Abstract
The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann--Neumann methods that have better convergence properties and require less computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear... (More)
The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann--Neumann methods that have better convergence properties and require less computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov--Poincaré operators. (Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Working paper/Preprint
publication status
published
subject
keywords
Nonoverlapping domain decomposition, Neumann–Neumann method, linear convergence, semi- and quasilinear elliptic equations
pages
22 pages
publisher
arXiv.org
DOI
10.48550/arXiv.2312.11177
project
Next generation numerical partitioning schemes for time dependent PDEs
language
English
LU publication?
yes
id
71f203be-93b0-4849-aae5-7b7f3ae95c49
date added to LUP
2023-12-19 08:47:44
date last changed
2024-01-18 15:42:17
@misc{71f203be-93b0-4849-aae5-7b7f3ae95c49,
  abstract     = {{The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann--Neumann methods that have better convergence properties and require less computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov--Poincaré operators.}},
  author       = {{Engström, Emil and Hansen, Eskil}},
  keywords     = {{Nonoverlapping domain decomposition; Neumann–Neumann method; linear convergence; semi- and quasilinear elliptic equations}},
  language     = {{eng}},
  month        = {{12}},
  note         = {{Preprint}},
  pages        = {{1--22}},
  publisher    = {{arXiv.org}},
  title        = {{Modified Neumann-Neumann methods for semi- and quasilinear elliptic equations}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2312.11177}},
  doi          = {{10.48550/arXiv.2312.11177}},
  year         = {{2023}},
}