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In this paper we are interested in the "fast path" fracture and we aim to useglobal-in-time, nonoverlapping domain decomposition methods to model flow andtransport problems in a porous medium containing such a fracture. We consider areduced model in which the fracture is treated as an interface between the twosubdomains. Two domain decomposition methods are considered: one uses thetime-dependent SteklovPoincar{\'e} operator and the other uses optimizedSchwarz waveform relaxation (OSWR) based on Ventcell transmission conditions.For each method, a mixed formulation of an interface problem on the space-timeinterface is derived, and different time grids are employed to adapt todifferent time scales in the subdomains and in the fracture. Demonstrations ofthe well-posedness of the Ventcell subdomain problems is given for the mixedformulation. An analysis for the convergence factor of the OSWR algorithm isgiven in the case with fractures to compute the optimized parameters. Numericalresults for two-dimensional problems with strong heterogeneities are presentedto illustrate the performance of the two methods.

Space-time Domain Decomposition and Mixed Formulation for solving reduced fracture models