  We construct a space-time parallel method for solving parabolic partial
differential equations by coupling the Parareal algorithm in time with
overlapping domain decomposition in space. The goal is to obtain a
discretization consisting of "local" problems that can be solved on parallel
computers efficiently. However, this introduces significant sources of error
that must be evaluated. Reformulating the original Parareal algorithm as a
variational method and implementing a finite element discretization in space
enables an adjoint-based a posteriori error analysis to be performed. Through
an appropriate choice of adjoint problems and residuals the error analysis
distinguishes between errors arising due to the temporal and spatial
discretizations, as well as between the errors arising due to incomplete
Parareal iterations and incomplete iterations of the domain decomposition
solver. We first develop an error analysis for the Parareal method applied to
parabolic partial differential equations, and then refine this analysis to the
case where the associated spatial problems are solved using overlapping domain
decomposition. These constitute our Time Parallel Algorithm (TPA) and
Space-Time Parallel Algorithm (STPA) respectively. Numerical experiments
demonstrate the accuracy of the estimator for both algorithms and the
iterations between distinct components of the error.
