Optimized Schwarz Methods with Overlap for the Helmholtz Equation

For the Helmholtz equation, simple absorbing conditions of the form ∂n − iω were proposed as transmission condition (TC) in Schwarz methods first without overlap in [4], and later also with overlap, see [3, 12]. More advanced TCs can also be used, see e.g. [11, 14, 2]. Furthermore, parameters can be introduced into TCs and then optimized for rapid convergence, which led to the so called optimized Schwarz methods, see e.g. [6, 13] for elliptic equations. Without overlap, the parameters involved in some zeroand second-order TCs for the Helmholtz equation have been optimized in [8, 7]. With overlap, preliminary numerical studies of the parameters have been presented in [5, 9]. In this paper, we present the asymptotic solutions of the corresponding optimization problems with small overlap. We also compare the optimized parameters with other choices based on convergence factors and actual iteration numbers. We finally test for the first time Taylor second-order absorbing conditions for domain decomposition with overlap in the Helmholtz case.