A note on the H¹-convergence of the overlapping Schwarz waveform relaxation method for the heat equation

The overlapping Schwarz waveform relaxation method is a parallel iterative method for solving time-dependent PDEs. Convergence of the method for the linear heat equation has been studied under infinity norm but it was unknown under the energy norm at the continuous level. The question is interesting for applications concerning fluxes or gradients of the solutions. In this work, we show that the energy norm of the errors of iterates is bounded by their infinity norm. Therefore, we give an affirmative answer to this question for the first time.