Optimized Schwarz methods with Ventcell transmission conditions for model problems with continuously variable coefficients

The Ventcell transmission conditions can enhance greatly the convergence of subdomain iterations using optimization-based parameters. However, such optimization can now only be performed for model problems with constant coefficients and thus forms a gap between the theoretical analysis and the applications where the model problems are seldom to be constant coefficients. In this paper, we close this gap by a rigorous analysis on the optimized Schwarz methods with Ventcell transmission conditions for a model problem where the reaction term varies parallel to the interface continuously. We obtain the asymptotically optimal transmission parameters by optimizing a convergence factor that is obtained using the technique of separation of variables other than the often-used Fourier transform, and then the corresponding convergence factor estimate follows. We at last validate the theoretical findings by numerical experiments.