Efficient iterative methods and solvers for FEM analysis

Abstract

The main topic of this thesis is the three-field formulation of Biot's model of poroelasticity and preconditioners to solve the equations of the model after finite element discretization. The three field-formulation that is used describes the state of the porous media using displacement, fluid flux and pressure. Using this formulation we introduce a classical formulation of the model, use implicit time discretization and introduce the weak formulation for the timestep problem. The main focus is on the preconditioners for the timestep problem. Several block diagonal preconditioners based on Schur complements are presented and analysed on the level of the weak formulation and on the matrix level. The same approach is then applied to Biot-Barenblatt model that generalizes Biot's model. The results on condition number estimates are illustrated by numerical experiments. The author's contributions are based on the coauthored articles that cover the construction of Schur complement preconditioners. This thesis generalizes them by taking most of the analysis to the weak formulation which provides clearer presentation and discretization-independent results.