Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations
We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion pr...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Sciendo
2021-05-01
|
| Series: | Moroccan Journal of Pure and Applied Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.2478/mjpaa-2021-0014 |
| _version_ | 1818755224116920320 |
|---|---|
| author | Gander Martin J. Halpern Laurence Hubert Florence Krell Stella |
| author_facet | Gander Martin J. Halpern Laurence Hubert Florence Krell Stella |
| author_sort | Gander Martin J. |
| collection | DOAJ |
| description | We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important. |
| first_indexed | 2024-12-18T05:35:45Z |
| format | Article |
| id | doaj.art-85df07134545443d9c1bcacb3b33ea1e |
| institution | Directory Open Access Journal |
| issn | 2351-8227 |
| language | English |
| last_indexed | 2024-12-18T05:35:45Z |
| publishDate | 2021-05-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Moroccan Journal of Pure and Applied Analysis |
| spelling | doaj.art-85df07134545443d9c1bcacb3b33ea1e2022-12-21T21:19:20ZengSciendoMoroccan Journal of Pure and Applied Analysis2351-82272021-05-017218221310.2478/mjpaa-2021-0014Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizationsGander Martin J.0Halpern Laurence1Hubert Florence2Krell Stella3University of Geneva, 2-4 rue du Lièvre, CP 64, 1211Genève, Switzerland.LAGA, Université Sorbonne Paris Nord et CNRS, 93430Villetaneuse, France.Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France, 39 rue F. Joliot Curie, 13453 Marseille, Cedex 13.Université de Nice, Parc Valrose, 28 avenue Valrose, 06108 Nice, Cedex 2, France.We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.https://doi.org/10.2478/mjpaa-2021-0014non-overlapping optimized schwarz methodventell transmission conditionddfv finite volume schemeanistropic diffusion65n5565n0835a3565f10 |
| spellingShingle | Gander Martin J. Halpern Laurence Hubert Florence Krell Stella Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations Moroccan Journal of Pure and Applied Analysis non-overlapping optimized schwarz method ventell transmission condition ddfv finite volume scheme anistropic diffusion 65n55 65n08 35a35 65f10 |
| title | Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations |
| title_full | Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations |
| title_fullStr | Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations |
| title_full_unstemmed | Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations |
| title_short | Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations |
| title_sort | optimized schwarz methods with general ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations |
| topic | non-overlapping optimized schwarz method ventell transmission condition ddfv finite volume scheme anistropic diffusion 65n55 65n08 35a35 65f10 |
| url | https://doi.org/10.2478/mjpaa-2021-0014 |
| work_keys_str_mv | AT gandermartinj optimizedschwarzmethodswithgeneralventcelltransmissionconditionsforfullyanisotropicdiffusionwithdiscretedualityfinitevolumediscretizations AT halpernlaurence optimizedschwarzmethodswithgeneralventcelltransmissionconditionsforfullyanisotropicdiffusionwithdiscretedualityfinitevolumediscretizations AT hubertflorence optimizedschwarzmethodswithgeneralventcelltransmissionconditionsforfullyanisotropicdiffusionwithdiscretedualityfinitevolumediscretizations AT krellstella optimizedschwarzmethodswithgeneralventcelltransmissionconditionsforfullyanisotropicdiffusionwithdiscretedualityfinitevolumediscretizations |