Asymptotic Results of Schwarz Waveform Relaxation Algorithm for Time Fractional Cable Equations

The equioscillation principle is an important rule to fix the parameter for the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions. For parabolic PDEs with integer order temporal derivative, such a principle yields optimal Robin parameter, while in our previous study we found numerically that it is not always the case for time fractional PDEs: the Robin parameter determined by the equioscillation principle is sometimes far away from optimal. In this paper, by using the time fractional Cable equations as the model, we show that our previous finding does not happen occasionally but an inherent property of the SWR algorithm. Our analysis also reveals an essential difference between the asymptotic convergence rates in the overlapping and non-overlapping cases. Numerical results are provided to validate our theoretical analysis.