The overset grid nicknamed "Yin-Yang" grid is singularity free and has quasi-uniform grid spacing. It is composed of two identical latitude/longitude orthogonal grid panels that are combined to cover the sphere with partial overlap on their boundaries. The system of shallow-water equations (SWEs) is a hyperbolic system at the core of many models of the atmosphere. In this paper, the SWEs are solved on the Yin-Yang grid by using an implicit and semi-Lagrangian discretization on a staggered mesh. The resulting scalar elliptic equation is solved using a Schwarz-type domain decomposition method, known as the optimized Schwarz method, which gives better performance than the classical Schwarz method by using specific Robin or higher order transmission conditions.

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Summary. We investigate the performance of domain decomposition methods for solving the Poisson equation on the surface of the sphere. This equation arises in a global weather model as a consequence of an implicit time discretization. We consider two different types of algorithms: the Dirichlet-Neumann algorithm and the optimal Schwarz method. We show that both algorithms applied to a simple two subdomain decomposition of the surface of the sphere converge in two iterations. While the Dirichlet-Neumann algorithm achieves this with local transmission conditions, the optimal Schwarz algorithm needs non-local transmission conditions. This seems to be a disadvantage of the optimal Schwarz method. We then show however that for more than two subdomains or overlapping subdomains, both the optimal Schwarz algorithm and the Dirichlet Neumann algorithm need non-local interface conditions to converge in a finite number of steps. Hence the apparent advantage of DirichletNeumann over optimal Schwarz is only an artifact of the special two subdomain decomposition.

Abstract. We investigate the performance of algebraic optimized Schwarz methods used as preconditioners for the solution of discretized differential equations. These methods consist on modifying the so-called transmission blocks. The transmission blocks are replaced by new blocks in order to improve the convergence of the corresponding iterative algorithms. In the optimal case, convergence in two iterations can be achieved. We are also interested in the behavior of the algebraic optimized Schwarz methods with respect to changes in the problems parameters. We focus on constructing preconditioners for different numerically challenging differential problems such as: Periodic and Torus problems; Meshfree problems; Three-dimensional problems. We present different numerical simulations corresponding to different type of problems in two-and three-dimensions.