Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

We perform a convergence analysis of a two-grid fixed stress split algorithm for the Biot system modeling coupled flow and deformation in heterogeneous poroelastic media. The two-grid fixed stress split scheme solves the flow subproblem on a fine grid using a multipoint flux mixed finite element method by imposing the fixed mean stress constraint followed by the poromechanics subproblem on a coarse grid using a conforming Galerkin method in every coupling iteration at each time step. 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 convergence 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 effective elastic properties for the coarse scale poromechanical solve in terms of the fine scale elastic properties. We analyze the contraction map with a numerical result comparing the numerically computed to the theoretically obtained contraction constant for a poroelastic medium with substantial spatial variability in the poroelastic moduli.