EFFICIENT DISCRETIZATION TECHNIQUES AND DOMAIN DECOMPOSITION METHODS FOR POROELASTICITY

This thesis develops a new mixed finite element method for linear elasticity model with weakly enforced symmetry on simplicial and quadrilateral grids. Motivated by the multipoint flux mixed finite element method (MFMFE) for flow in porous media, the method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces and the trapezoidal (vertex) quadrature rule in order to localize the interaction of degrees of freedom. Particularly, this allows for local elimination of stress and rotation variables around each vertex and leads to a cell-centered system for the displacements. The stability analysis shows that the method is well-posed on simplicial and quadrilateral grids. Theoretical and numerical results indicate first-order convergence for all variables in the natural norms. Further discussion of the application of said Multipoint Stress Mixed Finite Element (MSMFE) method to the Biot system for poroelasticity is then presented. The flow part of the proposed model is treated in the MFMFE framework, while the mixed formulation for the elasticity equation is adopted for the use of the MSMFE technique. The extension of the MFMFE method to an arbitrary order finite volume scheme for solving elliptic problems on quadrilateral and hexahedral grids that reduce the underlying mixed finite element method to cell-centered pressure system is also discussed. A Multiscale Mortar Mixed Finite Element method for the linear elasticity on non-matching multiblock grids is also studied. A mortar finite element space is introduced on the nonmatching interfaces. In this mortar space the trace of the displacement is approximated, and continuity of normal stress is then weakly imposed. The condition number of the interface system is analyzed and optimal order of convergence is shown for stress, displacement, and rotation. Moreover, at cell centers, superconvergence is proven for the displacement variable. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory for all proposed approaches