  This paper is concerned with global-in-time, nonoverlapping domain
decomposition methods for the mixed formulation of the diffusion problem. Two
approaches are considered: one uses the time-dependent Steklov-Poincar\'e
operator and the other uses Optimized Schwarz Waveform Relaxation (OSWR) based
on Robin transmission conditions. For each method, a mixed formulation of an
interface problem on the space-time interfaces between subdomains is derived,
and different time grids are employed to adapt to different time scales in the
subdomains. Demonstrations of the well-posedness of the subdomain problems
involved in each method and a convergence proof of the OSWR algorithm are given
for the mixed formulation. Numerical results for 2D problems with strong
heterogeneities are presented to illustrate the performance of the two methods.
