Optimized overlapping Schwarz methods for parabolic PDEs with time-delay

Abstract

We present overlapping Schwarz methods for the numerical solution of two model problems of delay PDEs: the heat equation with a fixed delay term, and the heat equation with a distributed delay in the form of an integral over the past. We first analyze properties of the solutions of these PDEs and find that their dynamics is fundamentally different from that of regular time-dependent PDEs without time delay. We then introduce and study overlapping Schwarz methods of waveform relaxation type for the two model problems. These methods compute the local solution in each subdomain over many time-levels before exchanging interface information to neighboring subdomains. We analyze the effect of the overlap and derive optimized transmission conditions of Robin type. Finally we illustrate the theoretical results and convergence estimates with numerical experiments.