Space-time domain decomposition methods for mixed formulations of flow and transport problems in porous media

Flow and transport problems in porous media are well-known for their high computational cost. In the simulation of an underground nuclear waste disposal site, one has to work with extremely different
length and time scales, and highly variable coefficients while satisfying strict accuracy requirements. One strategy for tackling these difficulties is to apply a non-overlapping domain decomposition method which allows local adaptation in both space and time and makes possible the use of parallel algorithms. We present two approaches. The first method uses a time-dependent Steklov-Poincaré operator and a generalized Neumann-Neumann
preconditioner with weight matrices to handle the heterogeneities. The second method uses the optimized Schwarz waveform relaxation (OSWR). The OSWR algorithm uses more general (Robin or Ventcell) transmission operators in which coefficients can be optimized to improve convergence rates. For both approaches, an interface problem is derived on the space-time interface. Thus, different time steps can be used in different subdomains adapted
to their physical properties. We use operator splitting to treat differently the advection and the diffunion. We show numerical experiments for various test cases, both academic and more realistic prototypes for nuclear waste disposal simulation. We finally study extensions of the two methods to the case in which the interface
represents a discrete-fracture in a reduced fracture model for flow in a fractured porous medium.