Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations
Engström, Emil LU and Hansen, Eskil LU
(2024)
In BIT Numerical Mathematics
64(4).
- 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... (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)
- author
- Engström, Emil
LU
and Hansen, Eskil
LU
- organization
- publishing date
- 2024-12
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- 35K20, 65J08, 65M55, Linear convergence, Nonoverlapping domain decompositions, Quasilinear parabolic equations, Space-time finite elements, Time-dependent Steklov–Poincaré operators
- in
- BIT Numerical Mathematics
- volume
- 64
- issue
- 4
- article number
- 37
- publisher
- Springer
- external identifiers
-
- scopus:85204883474
- ISSN
- 0006-3835
- DOI
- 10.1007/s10543-024-01038-5
- project
- Next generation numerical partitioning schemes for time dependent PDEs
- language
- English
- LU publication?
- yes
- id
- a0a4bf8a-fc47-4c8c-b249-c06a36077428
- date added to LUP
- 2025-01-09 13:12:19
- date last changed
- 2025-04-04 15:05:21
@article{a0a4bf8a-fc47-4c8c-b249-c06a36077428,
abstract = {{<p>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.</p>}},
author = {{Engström, Emil and Hansen, Eskil}},
issn = {{0006-3835}},
keywords = {{35K20; 65J08; 65M55; Linear convergence; Nonoverlapping domain decompositions; Quasilinear parabolic equations; Space-time finite elements; Time-dependent Steklov–Poincaré operators}},
language = {{eng}},
number = {{4}},
publisher = {{Springer}},
series = {{BIT Numerical Mathematics}},
title = {{Linearly convergent nonoverlapping domain decomposition methods for quasilinear parabolic equations}},
url = {{http://dx.doi.org/10.1007/s10543-024-01038-5}},
doi = {{10.1007/s10543-024-01038-5}},
volume = {{64}},
year = {{2024}},
}