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Convergence analysis of coupling iterations for the unsteady transmission problem with mixed discretizations

Monge, Azahar LU and Birken, Philipp LU (2016) 7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016 In ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering 1. p.1530-1544
Abstract

We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized one dimensional unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping intervals. The Laplacian is discretized using finite differences on one interval and finite elements on the other and the implicit Euler method is used for the time discretization. Following previous analysis where finite elements where used on both subdomains, we provide an exact formula for the spectral radius of the iteration matrix for this specific mixed discretizations. We then show that these tend to the ratio of heat conductivities in the semidiscrete spatial limit, but to a factor of the... (More)

We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized one dimensional unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping intervals. The Laplacian is discretized using finite differences on one interval and finite elements on the other and the implicit Euler method is used for the time discretization. Following previous analysis where finite elements where used on both subdomains, we provide an exact formula for the spectral radius of the iteration matrix for this specific mixed discretizations. We then show that these tend to the ratio of heat conductivities in the semidiscrete spatial limit, but to a factor of the ratio of the products of density and specific heat capacity in the semidiscrete temporal one. In the previous finite element analysis, the same result was obtained in the semidiscrete spatial limit but the factor in the temporal limit was lower. This explains the fast convergence previously observed for cases with strong jumps in the material coefficients. Numerical results confirm the analysis.

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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Coupled problems, Dirichlet-Neumann iteration, Fixed point iteration, Thermal fluid structure interaction, Transmission problem
in
ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering
volume
1
pages
15 pages
publisher
National Technical University of Athens
conference name
7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016
external identifiers
  • scopus:84995467062
ISBN
9786188284401
language
English
LU publication?
yes
id
f6f5529f-8498-40de-8241-25e596c872d5
date added to LUP
2017-04-24 15:20:19
date last changed
2017-04-24 15:20:36
@inproceedings{f6f5529f-8498-40de-8241-25e596c872d5,
  abstract     = {<p>We analyze the convergence rate of the Dirichlet-Neumann iteration for the fully discretized one dimensional unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping intervals. The Laplacian is discretized using finite differences on one interval and finite elements on the other and the implicit Euler method is used for the time discretization. Following previous analysis where finite elements where used on both subdomains, we provide an exact formula for the spectral radius of the iteration matrix for this specific mixed discretizations. We then show that these tend to the ratio of heat conductivities in the semidiscrete spatial limit, but to a factor of the ratio of the products of density and specific heat capacity in the semidiscrete temporal one. In the previous finite element analysis, the same result was obtained in the semidiscrete spatial limit but the factor in the temporal limit was lower. This explains the fast convergence previously observed for cases with strong jumps in the material coefficients. Numerical results confirm the analysis.</p>},
  author       = {Monge, Azahar and Birken, Philipp},
  booktitle    = {ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering},
  isbn         = {9786188284401},
  keyword      = {Coupled problems,Dirichlet-Neumann iteration,Fixed point iteration,Thermal fluid structure interaction,Transmission problem},
  language     = {eng},
  pages        = {1530--1544},
  publisher    = {National Technical University of Athens},
  title        = {Convergence analysis of coupling iterations for the unsteady transmission problem with mixed discretizations},
  volume       = {1},
  year         = {2016},
}