Convergence analysis of the overlapping Schwarz waveform relaxation algorithm for reaction-diffusion equations with time delay

In this paper we study convergence of the overlapping Schwarz waveform relaxation (OSWR) algorithm for reaction-diffusion equations with time delay. We first prove linear convergence of the algorithm in the continuous case on infinite time intervals at a rate depending on the size of the overlaps, the time-delay argument and the coefficients of the equations. For the special case, i.e., the heat equation with a fixed time delay, the convergence rates of the OSWR algorithm presented in this paper are sharper than the existing results. We then prove that the linear convergence remains valid after spatial discretization and the convergence rates are robust with respect to mesh refinement. The convergence behaviour of the algorithm with an arbitrary number of subdomains is also investigated and it is shown that the convergence rate deteriorates with as number of subdomains increases and ameliorates as the overlap size increases.