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<p>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.</p>
</abstract>

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.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

LAGA, Université Sorbonne Paris Nord et CNRS

Aix Marseille Univ, CNRS, Centrale Marseille

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the...

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Published Online: Dec 28, 2020

Page range: 182 - 213

Received: Feb 16, 2020

Accepted: Nov 25, 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0014

Keywords
Non-overlapping optimized Schwarz method, Ventell transmission condition, DDFV finite volume scheme, Anistropic diffusion

© 2020 Martin J. Gander et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Published Online: Dec 28, 2020

Page range: 182 - 213

Received: Feb 16, 2020

Accepted: Nov 25, 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0014

KeywordsNon-overlapping optimized Schwarz method, Ventell transmission condition, DDFV finite volume scheme, Anistropic diffusion

© 2020 Martin J. Gander et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Non-overlapping optimized Schwarz method, Ventell transmission condition, DDFV finite volume scheme, Anistropic diffusion