Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Abstract

We analyze an optimized artificial fixed-stress iterative scheme for a space–time finite element approximation of the Biot system modeling fluid flow in deformable porous media. The iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step. The optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations. The separated subproblems of fluid flow, written as a mixed first order in space system, and mechanical deformation are discretized by space–time finite element methods of arbitrary order. Continuous and discontinuous Galerkin discretizations of the time variable are encountered. The convergence of the iterative schemes is proved for the continuous and fully discrete case. The choice of the optimization parameter is identified in the proofs of convergence of the iterations. The analyses are illustrated and confirmed by numerical experiments.