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This work serves as a primer to our efforts in arriving at convergence estimates for the fixed stress split iterative scheme for single phase flow coupled with small strain anisotropic poroelastoplasticity. The fixed stress split iterative scheme solves the flow subproblem with stress tensor fixed using a mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step.The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the iterative scheme is contractive.

We perform a convergence analysis of a two-grid staggered solution algorithm for the Biot system modeling coupled flow and deformation in heterogeneous poroelastic media. The algorithm first solves the flow subproblem on a fine grid using a mixed finite element method (by freezing a certain measure of the mean stress) followed by the poromechanics subproblem on a coarse grid using a conforming Galerkin method. Restriction operators map the fine scale flow solution to the coarse scale poromechanical grid and prolongation operators map the coarse scale poromechanical solution to the fine scale flow grid. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The analysis is based on studying the equations satisfied by the difference of iterates to show that the two-grid scheme is a contraction map under certain conditions. Those conditions are used to construct the restriction and prolongation operators as well as arrive at coarse scale elastic properties in terms of the fine scale data. We show that the adjustable parameter in the measure of the mean stress is linked to the Voigt and Reuss bounds frequently encountered in computational homogenization of multiphase composites.

We consider a poro-elastic region embedded into an elastic non-porous region. The elastic displacement equations are discretized by a continuous Galerkin scheme, while the flow equations for the pressure in the poro-elastic medium are discretized by either a continuous Galerkin scheme or a mixed scheme. Since the overall system of equations is very large, a fixed-stress algorithm is used at each time step to decouple the displacement from the flow equations in the poro-elastic region. We prove a priori error estimates for the resulting Galerkin scheme as well as the mixed scheme, with the expected order of accuracy, provided the algorithm is sufficiently iterated at each time step. These theoretical results are confirmed by a numerical experiment performed with the mixed scheme. A complete analysis including a posteriori estimates for both the Galerkin and the mixed scheme has been done but is too long to appear here.