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

TL;DR

This paper develops and analyzes an optimized iterative scheme for simulating fluid flow in deformable porous media using space-time finite element methods, improving convergence speed through parameter tuning.

Contribution

It introduces an optimized fixed-stress iteration scheme with adaptive stabilization for Biot system simulation, with proven convergence for continuous and discrete cases.

Findings

Convergence of the iteration scheme is proven theoretically.

Numerical experiments confirm the effectiveness of the optimization.

The method accelerates convergence compared to non-optimized schemes.

Abstract

In this work we analyze an optimized artificial fixed-stress iteration scheme for the numerical approximation of the Biot system modelling 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 discretizations of the time variable are encountered. The convergence of the iteration 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.