Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions

Ladislav Foltyn; Dalibor Lukáš; Ivo Peterek

Applications of Mathematics (2020)

  • Volume: 65, Issue: 2, page 173-190
  • ISSN: 0862-7940

Abstract

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We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At each time slice a pseudo-stationary elliptic heat equation is solved by means of a domain decomposition method (DDM). In the 2 d , case we employ a nonoverlapping Schur complement method, while in the 1 d case an overlapping Schwarz DDM is employed. We document computational efficiency, as well as theoretical convergence rates of FEM semi-discretization schemes on numerical examples.

How to cite

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Foltyn, Ladislav, Lukáš, Dalibor, and Peterek, Ivo. "Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions." Applications of Mathematics 65.2 (2020): 173-190. <http://eudml.org/doc/297241>.

@article{Foltyn2020,
abstract = {We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At each time slice a pseudo-stationary elliptic heat equation is solved by means of a domain decomposition method (DDM). In the $2d$, case we employ a nonoverlapping Schur complement method, while in the $1d$ case an overlapping Schwarz DDM is employed. We document computational efficiency, as well as theoretical convergence rates of FEM semi-discretization schemes on numerical examples.},
author = {Foltyn, Ladislav, Lukáš, Dalibor, Peterek, Ivo},
journal = {Applications of Mathematics},
keywords = {domain decomposition method; parareal method; finite element method; heat equation},
language = {eng},
number = {2},
pages = {173-190},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions},
url = {http://eudml.org/doc/297241},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Foltyn, Ladislav
AU - Lukáš, Dalibor
AU - Peterek, Ivo
TI - Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 173
EP - 190
AB - We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At each time slice a pseudo-stationary elliptic heat equation is solved by means of a domain decomposition method (DDM). In the $2d$, case we employ a nonoverlapping Schur complement method, while in the $1d$ case an overlapping Schwarz DDM is employed. We document computational efficiency, as well as theoretical convergence rates of FEM semi-discretization schemes on numerical examples.
LA - eng
KW - domain decomposition method; parareal method; finite element method; heat equation
UR - http://eudml.org/doc/297241
ER -

References

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