Convergence analysis of coupling iterations for the unsteady transmission problem with mixed discretizations
(2016) 7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016 1. p.15301544 Abstract
We analyze the convergence rate of the DirichletNeumann 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 DirichletNeumann 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
 Monge, Azahar ^{LU} and Birken, Philipp ^{LU}
 organization
 publishing date
 2016
 type
 Chapter in Book/Report/Conference proceeding
 publication status
 published
 subject
 keywords
 Coupled problems, DirichletNeumann iteration, Fixed point iteration, Thermal fluid structure interaction, Transmission problem
 host publication
 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
 conference location
 Crete, Greece
 conference dates
 20160605  20160610
 external identifiers

 scopus:84995467062
 ISBN
 9786188284401
 language
 English
 LU publication?
 yes
 id
 f6f5529f849840de824125e596c872d5
 date added to LUP
 20170424 15:20:19
 date last changed
 20220130 19:45:48
@inproceedings{f6f5529f849840de824125e596c872d5, abstract = {{<p>We analyze the convergence rate of the DirichletNeumann 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}}, keywords = {{Coupled problems; DirichletNeumann iteration; Fixed point iteration; Thermal fluid structure interaction; Transmission problem}}, language = {{eng}}, pages = {{15301544}}, publisher = {{National Technical University of Athens}}, title = {{Convergence analysis of coupling iterations for the unsteady transmission problem with mixed discretizations}}, volume = {{1}}, year = {{2016}}, }