Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions

Optimized Schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions. We study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel (higher order) type, which can also be used in the various limits for example for the heat equation, since we include in our analysis a positive low frequency limiter both in space and time. We give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms, and which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, including cases not covered by our analysis, for example decompositions with cross points. We use Q1 finite element discretizations in space and Forward and Backward Euler discretizations in time (other discretization could also have been considered, since all our analysis is at the continuous level), and in all cases, we measure performance corresponding to our analysis.