# A Discontinuous Coarse Space (DCS) Algorithm for Cell Centered Finite Volume Based Domain Decomposition Methods: The DCS-RJMin Algorithm

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

## Abstract

In this paper, we introduce a new coarse space algorithm, the “Discontinuous Coarse Space Robin Jump Minimizer” (DCS-RJMin), to be used in conjunction with one-level domain decomposition methods (DDMs). This new algorithm makes use of Discontinuous Coarse Spaces (DCS), and is designed for DDM that naturally produce discontinuous iterates such as Optimized Schwarzs Methods (OSM). This algorithm is suitable both at the continuous level and for cell-centered finite volume discretizations. At the continuous level, we prove, under some conditions on the parameters of the algorithm, that the difference between two consecutive iterates goes to 0. We also provide numerical results illustrating the convergence behavior of the DCS-RJMin algorithm.

## References

1. 1.
R. Cautres, R. Herbin, F. Hubert, The Lions domain decomposition algorithm on non-matching cell centred finite volume meshes. IMA J. Numer. Anal. 24(3), 465–490 (2004)
2. 2.
B. Després, Domain decomposition method and the Helmholtz problem, in Mathematical and Numerical Aspects of Wave Propagation Phenomena, ed. by G.C. Cohen, L. Halpern, P. Joly. Proceedings in Applied Mathematics Series, vol. 50 (Society for Industrial and Applied Mathematics, Strasbourg, 1991), pp. 44–52Google Scholar
3. 3.
V. Dolean, F. Nataf, R. Scheichl, N. Spillane, Analysis of a two-level schwarz method with coarse spaces based on local dirichlet to neumann maps. Comput. Methods Appl. Math. 12(4), 391–414 (2012)
4. 4.
M. Dryja, O.B. Widlund, An additive variant of the Schwarz alternating method for the case of many subregions. Technical Report 339, also Ultracomputer Note 131, Department of Computer Science, Courant Institute (1987)Google Scholar
5. 5.
M. Dryja, O.B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Commun. Pure Appl. Math. 48(2), 121–155 (1995)
6. 6.
O. Dubois, Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients. Ph.D. thesis, McGill University, 2007Google Scholar
7. 7.
O. Dubois, M.J. Gander, Convergence behavior of a two-level optimized Schwarz preconditioner, in Domain Decomposition Methods in Science and Engineering, XVIII, ed. by M. Bercovier, M.J. Gander, R. Kornhuber, O. Widlund. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2009)Google Scholar
8. 8.
O. Dubois, M.J. Gander, S. Loisel, A. St-Cyr, D. Szyld, The optimized Schwarz method with a coarse grid correction. SIAM J. Sci. Comput. 34(1), A421–A458 (2012)
9. 9.
J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8(5), 1621–1644 (2010)
10. 10.
M.J. Gander, L. Halpern, K. Santugini Repiquet, A new coarse grid correction for RAS/AS, in Domain Decomposition Methods in Science and Engineering, XXI, ed. by J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, O. Widlund. Lecture Notes in Computational Science and Engineering (Springer, Switzerland, 2014), pp. 275–283Google Scholar
11. 11.
M.J. Gander, L. Halpern, K. Santugini Repiquet, Discontinuous coarse spaces for DD-methods with discontinuous iterates, in Domain Decomposition Methods in Science and Engineering XXI, ed. by J. Erhel, M.J. Gander, L. Halpern, G. Pichot, T. Sassi, O. Widlund. Lecture Notes in Computational Science and Engineering (Springer, Switzerland, 2014), pp. 607–615Google Scholar
12. 12.
M.J. Gander, F. Kwok, K. Santugini, Optimized Schwarz at cross points: finite volume case (2015, in preparation)Google Scholar
13. 13.
L. Halpern, F. Hubert, A finite volume Ventcell-Schwarz algorithm for advection-diffusion equations. SIAM J. Numer. Anal. 52(3), 1269–1291 (2014)
14. 14.
P.-L. Lions, On the Schwarz alternating method. III: a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, ed. by T.F. Chan, R. Glowinski, J. Périaux, O. Widlund (SIAM, Philadelphia, 1990), pp. 202–223Google Scholar
15. 15.
J. Mandel, Balancing domain decomposition. Commun. Numer. Methods Eng. 9(3), 233–241 (1993)
16. 16.
J. Mandel, M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients. Math. Comput. 65, 1387–1401 (1996)
17. 17.
J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers. Numer. Math. 73, 473–487 (1996)
18. 18.
F. Nataf, H. Xiang, V. Dolean, N. Spillane, A coarse sparse construction based on local Dirichlet-to-Neumann maps. SIAM J. Sci. Comput. 33(4), 1623–1642 (2011)
19. 19.
R.A. Nicolaides, Deflation conjugate gradients with application to boundary value problems. SIAM J. Numer. Anal. 24(2), 355–365 (1987)
20. 20.
K. Santugini, A discontinuous galerkin like coarse space correction for domain decomposition methods with continuous local spaces: the DCS-DGLC algorithm. ESAIM Proc. Surv. 45, 275–284 (2014)
21. 21.
N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2013) 