The Optimized Schwarz Method with a Coarse Grid Correction

Abstract

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.

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

“…This limitation has been successfully addressed for certain classes of problems, by adding a component to the algorithm that is known in the DDM community as a "coarse grid" [8,9,10,11,12]. This generic name refers to any technique that enables global sharing of information between subdomains, while the basic additive algorithm only allows local exchange of information, hence hampering convergence (note that multiplicative Schwarz algorithms enable long range exchange of information in one direction only, and cannot guarantee a convergence rate independent of the number of subdomains.)…”

“…This generic name refers to any technique that enables global sharing of information between subdomains, while the basic additive algorithm only allows local exchange of information, hence hampering convergence (note that multiplicative Schwarz algorithms enable long range exchange of information in one direction only, and cannot guarantee a convergence rate independent of the number of subdomains.) While coarse grids have proven to be very effective for Laplace-type problems, designing effective coarse grids for high-frequency Helmholtz problems proves difficult [11,12].…”

Discontinuous Coarse Spaces for DD-Methods with Discontinuous Iterates

“…Combining coarse spaces with methods with discontinuous iterates, such as optimized Schwarz methods (OSM [8]) is also non-trivial, see [6] and chapter 5 in [5] which contain extensive numerical tests, and [7] for a rigorous analysis of a special case. For restricted Additive Schwarz (RAS [1]), which also produces discontinuous global iterates since they are glued from local ones by the R operators in RAS in an aribirary fashion, see [9] in the present proceedings.…”

Nonlinear Schwarz iterations with reduced rank extrapolation

“…Such methods are well-known to provide very efficient preconditioners as well. Different boundary conditions on the artificial interfaces, such as Dirichlet or Robin conditions, have been developed; see, e.g., [7,8,18,20,29].Extrapolation methods are used for accelerating the convergence of large class of vector sequences; see, e.g., the review article by Smith, Ford and Sidi [28], or the book by Brezinski and Redivo-Zaglia [2]. In particular, these methods are employed to accelerate the convergence of fixed point iterative techniques for linear and nonlinear systems of equations; see, e.g [2,9,14,28].…”

“…Such methods are well-known to provide very efficient preconditioners as well. Different boundary conditions on the artificial interfaces, such as Dirichlet or Robin conditions, have been developed; see, e.g., [7,8,18,20,29].…”