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

In this paper, we consider a combined mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart-Thomas finite elements for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable. This numerical approach appears to be a common choice in the existing reservoir engineering simulators. We focus on deriving error estimates 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 convergence in space with respect to the L-2-norm for the pressure and for the average fluid velocity and also first order convergence in space with respect to the H1-norm for the displacement. Here we prove one extra order of convergence for the displacement approximation with respect to the L-2-norm. We also demonstrate that, by including a postprocessing step in the scheme, the order of convergence of the approximation of pressure can be improved. Even though this result is critical for deriving the L-2-norm 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.