The optimized Schwarz method and the closely related 2-Lagrange multiplier method are domain decomposition methods which can be used to parallelize the solution of partial differential equations. Although these methods are known to work well in special cases (e.g., when the domain is a square and the two subdomains are rectangles), the problem has never been systematically stated nor analyzed for general domains with general subdomains. The problem of cross points (when three or more subdomains meet at a single vertex) has been particularly vexing. We introduce a 2-Lagrange multiplier method for domain decompositions with cross points. We estimate the condition number of the iteration and provide an optimized Robin parameter for general domains. We hope that this new systematic theory will allow broader utilization of optimized Schwarz and 2-Lagrange multiplier preconditioners.

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.

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft. We describe a novel and efficient algorithm for calculating the field of values boundary, \partialW(\cdot), of an arbitrary complex square matrix: the boundary is described by a system of ordinary differential equations which are solved using Runge-Kutta (Dormand-Prince) numerical integration to obtain control points with derivatives, then finally Hermite interpolation is applied to produce a dense output. The algorithm computes \partialW(\cdot) both efficiently and with low error. Formal error bounds are proven for specific classes of matrix. Furthermore, we summarize the existing state of the art and make comparisons with the new algorithm. Finally, numerical experiments are performed to quantify the cost-error trade-off between the new algorithm and existing algorithms. \bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs. field of values, numerical range, Johnson's algorithm, Runge-Kutta, Dormand-Prince, parametrized eigenvalue problem, eigenvalue perturbation, eigenvalue crossing \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs .

Abstract. Optimized Schwarz Methods (OSM) use Robin transmission conditions across the subdomain interfaces. The Robin parameter can then be optimized to obtain the fastest convergence. A new formulation is presented with a coarse grid correction. The optimal parameter is computed for a model problem on a cylinder, together with the corresponding convergence factor which is smaller than that of classical Schwarz methods. A new coarse space is presented, suitable for OSM. Numerical experiments illustrating the effectiveness of OSM with a coarse grid correction, both as an iteration and as a preconditioner, are reported.1. Introduction. A popular and quite effective method for solving large elliptic problems is to subdivide the domain into many subdomains, and solve smaller elliptic problems on each subdomain in parallel. The Schwarz iteration reconciles these local solutions by using Dirichlet boundary conditions on the artificial interfaces between the subdomains and iterating; see, e.g., [39], [44]. These methods are also used as preconditioners. The idea of Optimized Schwarz Methods (OSM) is to use different boundary conditions on the artificial interfaces, such as Robin conditions, and take advantage of the fact that the Robin parameter can be optimized to obtain a faster convergence; see below for references.It is well-known that a coarse grid correction improves the convergence of the classical Schwarz methods, and in fact it is necessary in order to obtain weak scaling. An algorithm scales weakly if it can solve a larger problem in reasonable time by increasing the number of processors.

Abstract. Classical Schwarz methods and preconditioners subdivide the domain of a partial differential equation into subdomains and use Dirichlet transmission conditions at the artificial interfaces. Optimized Schwarz methods use Robin (or higher order) transmission conditions instead, and the Robin parameter can be optimized so that the resulting iterative method has an optimized convergence factor. The usual technique used to find the optimal parameter is Fourier analysis; but this is only applicable to certain regular domains, for example, a rectangle, and with constant coefficients. In this paper, we present a completely algebraic version of the optimized Schwarz method, including an algebraic approach to find the optimal operator or a sparse approximation thereof. This approach allows us to apply this method to any banded or block banded linear system of equations, and in particular to discretizations of partial differential equations in two and three dimensions on irregular domains. With the computable optimal operator, we prove that the optimized Schwarz method converges in no more than two iterations for the case of two subdomains. Similarly, we prove that when we use an optimized Schwarz preconditioner with this optimal parameter, the underlying minimal residual Krylov subspace method (e.g., GMRES) converges in no more than two iterations. Very fast convergence is attained even when the optimal transmission operator is approximated by a sparse matrix. Numerical examples illustrating these results are presented.