Regular Article

A priori error estimates for a discretized poro-elastic–elastic system solved by a fixed-stress algorithm

¹ Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France
² Center for Subsurface Modeling (CSM), Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712, USA
³ ARAMCO, Saudi Arabia

^* Corresponding author: girault@ann.jussieu.fr

Received: 8 March 2018
Accepted: 24 September 2018

Abstract

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.