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

This thesis contributes to the development of numerical methods for flow and transport in porous media, in particular, by using space-time domain decomposition methods that enable the use of different time steps in the sub-domains. In this work, we study two types of methods: one is based on a generalization of the Steklov-Poincare operator to time-dependent problems and one is based on the Optimized Schwarz Waveform Relaxation (OSWR) method in which more general (Robin or Ventcell) transmission conditions are used to accelerate the convergence of the method. These two methods are derived in a mixed formulation, which is well-suited to problems arising in the modeling of flow and transport in porous media. We first consider the diffusion problem and formulate an interface problem on the space-time interfaces between the sub-domains for each method. The well-posedness of the sub-domain problem with either Dirichlet or Robin boundary conditions is shown. The convergence proofs of the OSWR algorithm and of the semi-discrete OSWR algorithm in mixed form with nonconforming time discretization are given. Numerical experiments in 2D comparing the performance of the two methods for strongly heterogeneous problems are carried out with a time-dependent Neumann-Neumann pre-conditioner with weight matrices being used to accelerate the first method. We then extend both methods to the advection diffusion equation where operator splitting is used to treat the advection and the diffusion differently. Separate transmission conditions for the advection equation and for the diffusion equation are derived. Using numerical results for various test cases, both advection-dominated and diffusion-dominated problems, we compare the convergence of the two methods and analyze the accuracy in time given by each when nonconforming time grids are used. Two prototypes for nuclear waste disposal simulation are considered and time windows are used for long-term simulation. We also consider the OSWR method with Ventcell transmission conditions extended to the mixed formulation. The sub-domain problem with Ventcell boundary conditions is shown to be well-posed. We compare numerically, for a decomposition into two sub-domains, the performance of the optimized Ventcell and Robin parameters for heterogeneous problems. 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. (author)