Higher order variational time discretization and mixed finite element methods for flow in porous media

Numerical simulations of time dependent fluid flow in porous media are desirable in several fields of natural sciences and in a large number of branches of technology. However, the accurate numerical approximation of flow processes in heterogeneous porous media continues to be a challenging task. The applicability and value of the mixed finite element method and its hybrid variant have been demonstrated for a wide range of problem. While the discretization in space involves a significant set of complexiti es and challenges, the temporal approximations for transient flows in porous media have received relatively little interest and have been limited to traditional non-adaptive first and second order methods. In particular, rigorous studies of higher order t ime discretizations are missing. The objective of this project is to develop higher order temporal discretization of partial differential equations and describe, investigate theoretically and compare by means of numerical experiments the higher order Gal erkin time discretization applied to mixed finite element approximations. At first, this is done for a linear convection-diffusion equation as a mathematical prototype model. For some type of mixed methods the presence of a convective term is known to be essential and to cause a loss in the order of convergence. Then, we study flow in porous media modeled by the nonlinear Richards' equation which couples fluid mass conservation with Darcy's law. Higher order continuous Galerkin-Petrov and discontinuous Ga lerkin time discretizations are developed. Further, we study the potential and advantage of this uniform variational approach in space and time for the construction of simultaneous space-time adaptive methods. Finally, the numerical schemes will be applie d to relevant benchmarks to illustrate their efficiency.