Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/30a63444-ea35-4f95-93fa-70e72a8f0943
- author
- Engström, Emil LU and Hansen, Eskil LU
- organization
- publishing date
- 2022
- 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
- 2024-08-07 08:40:29
@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}}, }