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Domain decomposition methods for nonlinear elliptic and parabolic equations

Engström, Emil LU (2025)
Abstract
Nonoverlapping domain decomposition methods have been utilized for a long time to solve linear and nonlinear elliptic problems, and more recently, parabolic problems. Despite this, there is no convergence theory for nonlinear elliptic and parabolic equations on general Lipschitz domains in Rd, d ≥ 2. We therefore develop a Steklov–Poincaré theory for nonlinear elliptic and parabolic problems and study the properties of the Steklov–Poincaré operators. In the elliptic case, we show that the two standard methods, the Dirichlet–Neumann and Robin–Robin methods converge. We demonstrate with numerical results that the Neumann-Neumann method does not converge in some cases and instead develop two modified Neumann–Neumann methods and prove their... (More)
Nonoverlapping domain decomposition methods have been utilized for a long time to solve linear and nonlinear elliptic problems, and more recently, parabolic problems. Despite this, there is no convergence theory for nonlinear elliptic and parabolic equations on general Lipschitz domains in Rd, d ≥ 2. We therefore develop a Steklov–Poincaré theory for nonlinear elliptic and parabolic problems and study the properties of the Steklov–Poincaré operators. In the elliptic case, we show that the two standard methods, the Dirichlet–Neumann and Robin–Robin methods converge. We demonstrate with numerical results that the Neumann-Neumann method does not converge in some cases and instead develop two modified Neumann–Neumann methods and prove their convergence. In the parabolic case, we show convergence of the Robin–Robin method and introduce two modified Dirichlet–Neumann methods, for which we show convergence. We also discuss how the results can be applied to an equation after it has been discretized by a finite element method and give numerical results for each method. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Prof. Målqvist, Axel, University of Gothenburg, Sweden.
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Nonoverlapping domain decomposition, Steklov-Poincaré operator, convergence, Nonlinear elliptic equation, Nonlinear parabolic equation
pages
120 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Lecture Hall MH:Riesz, Centre of Mathematical Sciences, Sölvegatan 18 A, Faculty of Engineering LTH, Lund University, Lund.
defense date
2025-03-14 13:15:00
ISBN
978-91-8104-347-1
978-91-8104-348-8
project
Next generation numerical partitioning schemes for time dependent PDEs
Moving domain decomposition methods for parabolic PDEs
language
English
LU publication?
yes
id
45f3bc1d-f7e1-419d-b49c-b3ded28ca698
date added to LUP
2025-01-30 14:42:51
date last changed
2025-04-04 14:09:34
@phdthesis{45f3bc1d-f7e1-419d-b49c-b3ded28ca698,
  abstract     = {{Nonoverlapping domain decomposition methods have been utilized for a long time to solve linear and nonlinear elliptic problems, and more recently, parabolic problems. Despite this, there is no convergence theory for nonlinear elliptic and parabolic equations on general Lipschitz domains in Rd, d ≥ 2. We therefore develop a Steklov–Poincaré theory for nonlinear elliptic and parabolic problems and study the properties of the Steklov–Poincaré operators. In the elliptic case, we show that the two standard methods, the Dirichlet–Neumann and Robin–Robin methods converge. We demonstrate with numerical results that the Neumann-Neumann method does not converge in some cases and instead develop two modified Neumann–Neumann methods and prove their convergence. In the parabolic case, we show convergence of the Robin–Robin method and introduce two modified Dirichlet–Neumann methods, for which we show convergence. We also discuss how the results can be applied to an equation after it has been discretized by a finite element method and give numerical results for each method.}},
  author       = {{Engström, Emil}},
  isbn         = {{978-91-8104-347-1}},
  keywords     = {{Nonoverlapping domain decomposition; Steklov-Poincaré operator; convergence; Nonlinear elliptic equation; Nonlinear parabolic equation}},
  language     = {{eng}},
  month        = {{01}},
  publisher    = {{Centre for Mathematical Sciences, Lund University}},
  school       = {{Lund University}},
  title        = {{Domain decomposition methods for nonlinear elliptic and parabolic equations}},
  url          = {{https://lup.lub.lu.se/search/files/207412400/thesis_electronic.pdf}},
  year         = {{2025}},
}