A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model

We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the sensitivity of the model with respect to its parameters

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

Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods