Physics-oriented solvers for multicompartmental poromechanics

This thesis deals with generalizations of the classical consolidation model, which was developed by Maurice Anthony Biot in 1941 and further elaborated in his subsequent scientific publications for various underlying structural characteristics. The aim of poromechanics is to describe the behavior of a porous, elastic solid material saturated by one or more fluids. In addition to the application of the considered systems of partial differential equations in the field of geomechanics for modeling oil recovery, they have recently also been used to simulate biomechanical processes in human organs such as the kidneys, liver or brain. Due to the wide range of feasible parameter values over several orders of magnitude, both the analysis and the efficient numerical solving become challenging problems. First, we study the multiple-network model where the porous material is permeated by several fluid flows. For a stable discretization, we combine the implicit Euler method in time with hybridized discontinuous Galerkin methods in space, whereby the mass of the individual fluids is preserved. Then, we prove the parameter-robust well-posedness of this discrete problem utilizing an abstract framework for perturbed saddle point problems. We next consider the dynamic Biot problem in three-field formulation, which incorporates inertia effects and hence contains additional acceleration terms compared to the quasi-static model. By means of a variational discretization in space and time, we generate a piecewise polynomial approximation of the solution, which is continuous over the entire time interval. Here, the spatial discretization is realized by a conforming finite element approach. We perform a thorough convergence analysis in which optimal error estimates in the energy variables are derived by aptly choosing test functions and applying the discrete Gronwall inequality. To guarantee strong mass conservation of the fluid for the dynamic Biot model also on the discrete level, we employ a discontinuous Galerkin approximation for the displacement field. In the analysis, we adapt the proof to the modified discretization and finally verify analogous error estimates. All theoretical results in this work are complemented by numerical experiments.