Accuracy of a coupled mixed and Galerkin finite element approximation for poroelasticity

In this paper, we consider a coupling mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart-Thomas space for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable while we consider the backward Euler method for the time discretization. This numerical scheme appears to be one common approach applied to existing reservoir engineering simulators. Theoretical convergence error estimates are derived in a discrete-in-time setting. Previous a priori error estimates described in the literature e.g. [2][19], which are optimal, show first order convergency with respect to the L-norm for the pressure and for the average fluid velocity approximation errors and with respect to the H-norm for the displacement approximation error. Here we prove one extra order of convergence for the displacement approximation with respect to the L-norm. We also demonstrate that, by including a post-processing step in the scheme, the order of convergence for the approximation of pressure can be improved. Even though this result is critical for deriving the Lnorm error estimates for the approximation of the deformation variable, surprisingly the corresponding gain of one convergence order holds independently of including or not the post-processing step in the method.