  The paper is concerned with overlapping domain decomposition and exponential
time differencing for the diffusion equation discretized in space by
cell-centered finite differences. Two localized exponential time differencing
methods are proposed to solve the fully discrete problem: the first method is
based on Schwarz iteration applied at each time step and involves solving
stationary problems in the subdomains at each iteration, while the second
method is based on the Schwarz waveform relaxation algorithm in which
time-dependent subdomain problems are solved at each iteration. The convergence
of the associated iterative solutions to the corresponding fully discrete
multidomain solution and to the exact semi-discrete solution is rigorously
proved. Numerical experiments are carried out to confirm theoretical results
and to compare the performance of the two methods.
