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Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations

Hansen, Eskil LU and Engström, Emil LU (2023)
Abstract
We prove linear convergence for a new family of modified Dirichlet–Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin–Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov–Poincaré operators. The convergence analysis... (More)
We prove linear convergence for a new family of modified Dirichlet–Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin–Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov–Poincaré operators. The convergence analysis is conducted without assuming restrictive regularity assumptions on the solutions or the numerical iterates. We also prove that these continuous convergence results extend to the discrete case obtained when combining domain decompositions with space-time finite elements. (Less)
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author
and
organization
publishing date
type
Working paper/Preprint
publication status
published
subject
pages
31 pages
publisher
arXiv.org
DOI
10.48550/arXiv.2308.15314
project
Next generation numerical partitioning schemes for time dependent PDEs
language
English
LU publication?
yes
id
afeeb801-fd2b-456e-b781-ac0dc861b396
date added to LUP
2023-12-06 16:04:20
date last changed
2024-01-18 10:10:53
@misc{afeeb801-fd2b-456e-b781-ac0dc861b396,
  abstract     = {{We prove linear convergence for a new family of modified Dirichlet–Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin–Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov–Poincaré operators. The convergence analysis is conducted without assuming restrictive regularity assumptions on the solutions or the numerical iterates. We also prove that these continuous convergence results extend to the discrete case obtained when combining domain decompositions with space-time finite elements.}},
  author       = {{Hansen, Eskil and Engström, Emil}},
  language     = {{eng}},
  note         = {{Preprint}},
  publisher    = {{arXiv.org}},
  title        = {{Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2308.15314}},
  doi          = {{10.48550/arXiv.2308.15314}},
  year         = {{2023}},
}