A new treatment of cross points in Optimized Schwarz Methods
The Optimized Schwarz Method (OSM) is a well established domain decomposition (DDM) strategy for solving frequency domain wave propagation problems such as Helmholtz equation.
In this method, the wave equation is solved independently in each subdomain imposing impedance conditions at the boundary. Coupling between subdomains is obtained via an exchange operator that swaps traces on each side of each interface. Whenever the subdomain partition does not involve any junction i.e. point where at least three subdomains abut, this strategy can be very efficient provided that the impedance of local subproblems is chosen wisely.
The situation is different when there are junctions and the presence of such points can spoil the convergence of the method, even for common geometric configurations.
The treatment of junctions in OSM has been the subject of many contributions and, although convincing numerical remedies are now available in the case of right-angled junctions, no generic satisfactory approach has been proposed so far.
In this talk we will present a new variant of OSM that can accomodate the presence of cross points of any shape. It is based on a modified exchange operator that requires, at each iteration of the linear solver, the computation of an orthogonal projection (in the sense of a well chosen scalar product) on the space of traces that match through interfaces i.e. the discrete Dirichlet single-trace space. The orthogonality of this exchange operator allows to establish a convergence result for our method that is uniform with respect to mesh resolution. Besides a detailed description of this method, we will present numerical result in 2D.
This is a joint work with E.Parolin (INRIA POems) supported by the project NonlocalDD from the French National Research Agency (ANR) through grant ANR-15-CE23-0017-01.