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

Engström, Emil LU and Hansen, Eskil LU orcid (2024) In Applied Numerical Mathematics 206. p.322-339
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 fewer 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 fewer 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.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Linear convergence, Neumann–Neumann method, Nonoverlapping domain decomposition, Semi- and quasilinear elliptic equations
in
Applied Numerical Mathematics
volume
206
pages
18 pages
publisher
Elsevier
external identifiers
  • scopus:85202193901
ISSN
0168-9274
DOI
10.1016/j.apnum.2024.08.011
project
Next generation numerical partitioning schemes for time dependent PDEs
language
English
LU publication?
yes
id
9b4fcd9f-4265-4c1f-9f64-4597a46c1a97
date added to LUP
2024-10-30 14:09:18
date last changed
2025-04-04 14:57:00
@article{9b4fcd9f-4265-4c1f-9f64-4597a46c1a97,
  abstract     = {{<p>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 fewer 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.</p>}},
  author       = {{Engström, Emil and Hansen, Eskil}},
  issn         = {{0168-9274}},
  keywords     = {{Linear convergence; Neumann–Neumann method; Nonoverlapping domain decomposition; Semi- and quasilinear elliptic equations}},
  language     = {{eng}},
  pages        = {{322--339}},
  publisher    = {{Elsevier}},
  series       = {{Applied Numerical Mathematics}},
  title        = {{Modified Neumann–Neumann methods for semi- and quasilinear elliptic equations}},
  url          = {{http://dx.doi.org/10.1016/j.apnum.2024.08.011}},
  doi          = {{10.1016/j.apnum.2024.08.011}},
  volume       = {{206}},
  year         = {{2024}},
}