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Convergence analysis of domain decomposition methods : Nonlinear elliptic and linear parabolic equations

Engström, Emil LU (2023) In Licentiate Theses in Mathematical Sciences 2023(2).
Abstract
Domain decomposition methods are widely used tools for solving partial differential equations in parallel. However, despite their long history, there is a lack of rigorous convergence theory for equations with non-symmetric differential operators. This includes both nonlinear elliptic equations and linear parabolic equations. The aim of this thesis is therefore twofold: First, to construct frameworks, based on new Steklov--Poincaré operators, that allow the study of nonoverlapping domain decomposition methods for nonlinear elliptic and linear parabolic equations. Second, to prove convergence of the Robin--Robin method using these frameworks. For the nonlinear elliptic case, this involves studying $L^p$-variants of the Lions--Magenes space.... (More)
Domain decomposition methods are widely used tools for solving partial differential equations in parallel. However, despite their long history, there is a lack of rigorous convergence theory for equations with non-symmetric differential operators. This includes both nonlinear elliptic equations and linear parabolic equations. The aim of this thesis is therefore twofold: First, to construct frameworks, based on new Steklov--Poincaré operators, that allow the study of nonoverlapping domain decomposition methods for nonlinear elliptic and linear parabolic equations. Second, to prove convergence of the Robin--Robin method using these frameworks. For the nonlinear elliptic case, this involves studying $L^p$-variants of the Lions--Magenes space. In the parabolic case, we use a variational formulation based on fractional time-regularity. The analysis is performed with weak requirements on the spatial domain, where we only assume that the domains have Lipschitz regularity, and for the solutions to the equations, where we assume that their normal derivatives over the interface is in $L^2(\Gamma)$. (Less)
Please use this url to cite or link to this publication:
author
supervisor
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Nonoverlapping domain decomposition, Steklov--Poincaré operator, Robin--Robin method, Convergence, Nonlinear elliptic equation, Space-time
in
Licentiate Theses in Mathematical Sciences
volume
2023
issue
2
pages
63 pages
publisher
Lund University
ISSN
1404-028X
ISBN
978-91-8039-579-3
978-91-8039-578-6
project
Next generation numerical partitioning schemes for time dependent PDEs
language
English
LU publication?
yes
id
7fb80724-2a99-42dd-bf3e-8573e51c4562
date added to LUP
2023-03-03 18:15:29
date last changed
2023-09-06 09:37:03
@misc{7fb80724-2a99-42dd-bf3e-8573e51c4562,
  abstract     = {{Domain decomposition methods are widely used tools for solving partial differential equations in parallel. However, despite their long history, there is a lack of rigorous convergence theory for equations with non-symmetric differential operators. This includes both nonlinear elliptic equations and linear parabolic equations. The aim of this thesis is therefore twofold: First, to construct frameworks, based on new Steklov--Poincaré operators, that allow the study of nonoverlapping domain decomposition methods for nonlinear elliptic and linear parabolic equations. Second, to prove convergence of the Robin--Robin method using these frameworks. For the nonlinear elliptic case, this involves studying $L^p$-variants of the Lions--Magenes space. In the parabolic case, we use a variational formulation based on fractional time-regularity. The analysis is performed with weak requirements on the spatial domain, where we only assume that the domains have Lipschitz regularity, and for the solutions to the equations, where we assume that their normal derivatives over the interface is in $L^2(\Gamma)$.}},
  author       = {{Engström, Emil}},
  isbn         = {{978-91-8039-579-3}},
  issn         = {{1404-028X}},
  keywords     = {{Nonoverlapping domain decomposition; Steklov--Poincaré operator; Robin--Robin method; Convergence; Nonlinear elliptic equation; Space-time}},
  language     = {{eng}},
  month        = {{02}},
  note         = {{Licentiate Thesis}},
  number       = {{2}},
  publisher    = {{Lund University}},
  series       = {{Licentiate Theses in Mathematical Sciences}},
  title        = {{Convergence analysis of domain decomposition methods : Nonlinear elliptic and linear parabolic equations}},
  url          = {{https://lup.lub.lu.se/search/files/139535266/emil_lic_lucris.pdf}},
  volume       = {{2023}},
  year         = {{2023}},
}