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We perform a convergence analysis of a two-grid staggered solution algorithmfor the Biot system modeling coupled flow and deformation in heterogeneousporoelastic media. The algorithm first solves the flow subproblem on a finegrid using a mixed finite element method (by freezing a certain measure of themean stress) followed by the poromechanics subproblem on a coarse grid using aconforming Galerkin method. Restriction operators map the fine scale flowsolution to the coarse scale poromechanical grid and prolongation operators mapthe coarse scale poromechanical solution to the fine scale flow grid. Thecoupling iterations are repeated until convergence and Backward Euler isemployed for time marching. The analysis is based on studying the equationssatisfied by the difference of iterates to show that the two-grid scheme is acontraction map under certain conditions. Those conditions are used toconstruct the restriction and prolongation operators as well as arrive atcoarse scale elastic properties in terms of the fine scale data. We show thatthe adjustable parameter in the measure of the mean stress is linked to theVoigt and Reuss bounds frequently encountered in computational homogenizationof multiphase composites.

The correspondence between Voigt and Reuss bounds and the decoupling  constraint in a two-grid staggered solution algorithm to coupled flow and  deformation in heterogeneous poroelastic media