Optimized Schwarz Methods for Domains with an Arbitrary Interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate its convergence. In the literature, analysis of optimized Schwarz methods rely on Fourier analysis and so the domains are restricted to be regular (rectangle or disk). By expressing the interface operator in terms of Poincare–Steklov operators, we are able to derive upper bounds of the spectral radius of the operator for Poisson-like problems for two essentially arbitrary subdomains. For a first order (Robin) boundary operator, an optimal choice of the parameter in the boundary operator leads to an upper bound of 1 − O(h) of the spectral radius, where h is the discretization parameter. For a certain higher order boundary operator, a clever choice of the two parameters in the boundary operator leads to an upper bound of 1 − O(h ) of the spectral radius. These agree with the predicted rates for rectangular subdomains available in the literature and are also the observed rates in numerical simulations. This contribution summarizes the author’s work in [11, 12]. Let Ω be a bounded domain in IR with a smooth boundary. Suppose Ω is composed of two nonoverlapping open subdomains, that is, Ω = Ω 1 ∪ Ω2 with Ω1 ∩Ω2 = ∅. Assume that the artificial boundary Γ = Ω 1 ∩Ω2 is non-trivial (nonzero measure in RN−1) and is a smooth curve. We shall always assume that ∂Ωi \Γ is non-trivial for both i = 1, 2. Recall the trace space