Convergence analysis for non-iterative sequential schemes for Biot's model

TL;DR

This paper provides the first convergence analysis of an explicit fixed-stress split scheme for Biot's equations, demonstrating optimal convergence under certain finite element discretizations, and proposes a simplified decoupled algorithm.

Contribution

It offers the first theoretical convergence proof for the explicit fixed-stress split scheme in Biot's model and introduces a simplified, stabilized decoupled algorithm.

Findings

The explicit fixed-stress split scheme is optimally convergent with inf-sup stable finite elements.

A new stabilized decoupled algorithm for Biot's model is proposed and shown to be optimally convergent.

The analysis enhances understanding of explicit schemes for poroelasticity problems.

Abstract

An alternative to the fully implicit or monolithic methods used for the solution of the coupling of fluid flow and deformation in porous media is a sequential approach in which the fully coupled system is broken into subproblems (flow and mechanics problems) that are solved one after the other. This fully explicit coupling approach is a very simple scheme which allows much flexibility in the implementation and has a lower computational cost, making it quite attractive in practice since smaller linear systems need to be solved in order to obtain the solution for the whole coupled poroelastic system. Due to the appealing advantages of these methods, intensive research is currently being carried out in this direction, as in the present work. Although the application of this type of method is very common in practice, there exist only a few works devoted to their theoretical analysis. In this work, we consider the so-called explicit fixed-stress split scheme, which consists of solving the flow problem first with time-lagging the displacement term, followed by the solution of the mechanics problem. To the best of our knowledge, we provide the first convergence analysis of the explicit fixed-stress split scheme for Biot's equations. In particular, we prove that this algorithm is optimally convergent if the considered finite element discretization satisfies an inf-sup condition. In addition, with the aim of designing the simplest scheme for solving Biot's model, we also propose a similar decoupled algorithm for piecewise linear finite elements for both variables which arises from the novel stabilization recently proposed in \cite{Pe2025}, and is demonstrated to be optimally convergent.