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

• Published in: Moroccan journal of pure and applied analysis Vol. 7; no. 2; pp. 182 - 213
• Main Authors: Gander, Martin J., Halpern, Laurence, Hubert, Florence, Krell, Stella
• Format: Journal Article
• Language: English
• Published: Fez Sciendo 01.07.2021; De Gruyter Brill Sp. z o.o., Paradigm Publishing Services; Sidi Mohamed Benabdallah University [Ifrane] Al Akhawyn University
• Subjects: 35A35; 65F10; 65N08; 65N55; Anistropic diffusion; DDFV finite volume scheme; Mathematics; Non-overlapping optimized Schwarz method; Numerical Analysis; Ventell transmission condition; Ventell transmission condition; 65F10 Non-overlapping optimized Schwarz method; Anistropic diffusion; DDFV finite volume scheme; 65N08; 35A35; Mathematics Subject Classification . 65N55
• ISSN: 2351-8227; 2605-6364; 2351-8227
• DOI: 10.2478/mjpaa-2021-0014

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.