Optimized Discrete Schwarz Methods for Anisotropic Elliptic Problems (Morlet Chair – Martin Gander)

This research in pairs focuses on the development and the understanding of optimized discrete Schwarz methods based on the Discrete Duality Finite Volume (DDFV for short) discretization for anisotropic elliptic problems. The collaboration, based on the expertise of Martin Gander and Laurence Halpern on domain decomposition methods and of Florence Hubert and Stella Krell on DDFV methods, started about ten years ago.
Schwarz algorithms are a well-known strategy to solve problems on large domains iteratively, and they are often used at the discrete level as pre-conditioners for large linear systems [8, 3]. DDFV methods have been developed in the late 90’s [2, 1] to approximate linear and nonlinear elliptic problems on general meshes. The good properties of DDFV have extensively been tested in the benchmark [7]. Combining DDFV with Schwarz methods enabled us to propose new discrete algorithms for elliptic problems [6, 4]. We are focusing on Robin and Ventcell transmission conditions for non-overlapping Schwarz methods, on proving convergence of the algorithm, on the optimization of the transmission parameters, and comparing optimal parameters obtained for the continuous problem to the ones for the discrete problem on uniform grids and for the discrete problem on general grids [5].
​This research in pairs will enable us to complete our work including the hard problem of cross points (intersection of at least three domains), and to work on overlapping discrete Schwarz methods for anisotropic elliptic problems.

• Martin Gander (Université de Genève & Aix-Marseille Université)​
• Laurence Halpern (Université Sorbonne Paris Nord)
• Florence Hubert (Aix-Marseille Université)
• Stella Krell (Université Côte d’Azur)

[1] B. Andreianov, F. Boyer, and F. Hubert. Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D-meshes. Num. Meth. for PDEs, 23(1):145–195, 2007.

[2] K. Domelevo and P. Omnes. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal., 39(6):1203–1249, 2005.

[3] M. J. Gander. Schwarz methods over the course of time. Electronic transactions on numerical analysis, 31:228–255, 2008.

[4] M. J. Gander, L. Halpern, F. Hubert, and S. Krell. DDFV Ventcell Schwarz algorithms. In Domain Decomposition Methods in Science and Engineering XXII, pages 481–489. Springer, 2016.

[5] M. J. Gander, L. Halpern, F. Hubert, and S. Krell. Optimized Schwarz methods for anisotropic diffusion with discrete duality finite volume discretizations. submitted, 2019.

[6] M. J. Gander, F. Hubert, and S. Krell. Optimized Schwarz algorithms in the framework of ddfv schemes. In Domain Decomposition Methods in Science and Engineering XXI, pages 457–466. Springer, 2014.

[7] R. Herbin and F. Hubert. Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In R. Eymard and J.-M. Hérard, editors, Finite Volumes for Complex Applications V, pages 659– 692. John Wiley & Sons, 2008.

[8] P. L. Lions. On the Schwarz alternating method. III. A variant for non-overlapping subdomains. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), pages 202–223. SIAM, Philadelphia, PA, 1990.