One challenging task in the field of geomechanics is to compute the deformation of porous media saturated by an incompressible viscous fluid. This hydro-mechanical coupling is called poroelasticity and is modeled mathematically by a coupled partial differential equation of elliptic-parabolic type. For the numerical approximation of the displacement and the pressure variables one usually applies the implicit Euler scheme for the discretization in time and a Galerkin ansatz for the spatial discretization. Due to the coupling, this leads to very large linear systems, which need to be solved in every time step.
The aim of this project is to analyze convergence properties of semi-explicit methods applied to general linear elliptic-parabolic problems. This means that the implicit Euler scheme for the temporal discretization is replaced by an explicit-implicit version of it. As a consequence, the resulting linear systems decouple, which leads to a tremendous reduction of the computational complexity.
​Within this project, we also consider generalizations such as multiple-network poroelasticity models. Such problems appear, e.g., in medical applications and benefit even more from a possible decoupling. Further, we like to focus on elliptic-parabolic systems including heterogeneous coefficients. In this case, the semi-explicit discretization in time is combined with multiscale techniques in space.