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Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations

Engström, Emil LU and Hansen, Eskil LU (2022) In SIAM Journal on Numerical Analysis 60(2). p.585-605
Abstract
We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L^p-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the... (More)
We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L^p-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions. (Less)
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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Robin-Robin method, Nonoverlapping domain decomposition, Nonlinear elliptic equation, Convergence, Steklov-Poincaré operator
in
SIAM Journal on Numerical Analysis
volume
60
issue
2
pages
21 pages
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:85131248958
ISSN
0036-1429
DOI
10.1137/21M1414942
project
Next generation numerical partitioning schemes for time dependent PDEs
language
English
LU publication?
yes
id
30a63444-ea35-4f95-93fa-70e72a8f0943
date added to LUP
2021-04-26 18:46:39
date last changed
2023-03-03 11:32:03
@article{30a63444-ea35-4f95-93fa-70e72a8f0943,
  abstract     = {{We prove convergence for the nonoverlapping Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov-Poincaré operators based on the p-structure and the L^p-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.}},
  author       = {{Engström, Emil and Hansen, Eskil}},
  issn         = {{0036-1429}},
  keywords     = {{Robin-Robin method; Nonoverlapping domain decomposition; Nonlinear elliptic equation; Convergence; Steklov-Poincaré operator}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{585--605}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Numerical Analysis}},
  title        = {{Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations}},
  url          = {{https://lup.lub.lu.se/search/files/97125681/RR_nonlin.pdf}},
  doi          = {{10.1137/21M1414942}},
  volume       = {{60}},
  year         = {{2022}},
}