This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards' equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards' equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L−scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L−scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L−scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.

In this paper we describe the OpenGeoSys (OGS) project, which is a scientific open source initiative for numerical simulation of thermo-hydro-mechanical-chemical (THMC) processes in porous media. The basic concept is to provide a flexible numerical framework (using primarily the Finite Element Method (FEM)) for solving multi-field problems in porous and fractured media for applications in geoscience and hydrology. To this purpose OGS is based on an object-oriented FEM concept including a broad spectrum of interfaces for pre-and post-processing. The OGS idea has been in development since the mid eighties. We provide a short historical note about the continuous process of concept and software development having evolved through Fortran, C, and C++ implementations. The idea behind OGS is to provide an open platform to the community, outfitted with professional software engineering tools such as platform-independent compiling and automated benchmarking. A comprehensive benchmarking book has been prepared for publication. Benchmarking has been proven to be a valuable tool for cooperation between different developer teams, e.g. for code comparison and validation purposes (DEVOVALEX and CO2 BENCH projects). On one hand, object-orientation (OO) provides a suitable framework for distributed code development; however the parallelization of OO codes still lacks efficiency. High-performance-computin (HPC) efficiency of OO codes is subject to future research.

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Abstract. We analyze a discretization method for a class of degenerate parabolic problems that includes the Richards' equation. This analysis applies to the pressure-based formulation and considers both variably and fully saturated regimes. To overcome the difficulties posed by the lack in regularity, we first apply the Kirchhoff transformation and then integrate the resulting equation in time. We state a conformal and a mixed variational formulation and prove their equivalence. This will be the underlying idea of our technique to get error estimates.A regularization approach is combined with the Euler implicit scheme to achieve the time discretization. Again, equivalence between the two formulations is demonstrated for the semidiscrete case. The lowest order Raviart-Thomas mixed finite elements are employed for the discretization in space. Error estimates are obtained, showing that the scheme is convergent.

This thesis concerns iterative solvers for poromechanics problems. The problems in the studies have involved linear poromechanics, non-linear poromechanics, and poromechanics under large deformation. We included high order discretizations, applied linearization techniques and splitting methods to develop new solvers. We studied the robustness and convergence of these solvers. By studying the fixed stress method as an iterative solver for poromechanics, we developed an optimized version of it. Furthermore, by extending the convergence analysis in the time domain, we developed a new version of the fixed stress method that is partially parallelized. This splitting method was combined with linearization techniques to develop solvers for non-linear poromechanics. By studying the convergence of the linearisation schemes, we developed new solvers and extended the applicability to more complex phenomena, for instance poromechanics with large deformation.