A Robin-Robin splitting method for the Stokes-Biot fluid-poroelastic structure interaction model

TL;DR

This paper introduces a stable and efficient Robin-Robin splitting method for simulating fluid-poroelastic structure interactions, combining decoupled Stokes and Biot equations with proven convergence and stability.

Contribution

The paper develops a novel Robin-Robin splitting scheme for fluid-poroelastic interactions, providing stability analysis, error estimates, and iterative convergence proofs.

Findings

The method is unconditionally stable.

The time discretization error is $oxed{ ext{O}(\sqrt{T}\Delta t)}$.

Numerical experiments confirm theoretical results.

Abstract

We develop and analyze a splitting method for fluid-poroelastic structure interaction. The fluid is described using the Stokes equations and the poroelastic structure is described using the Biot equations. The transmission conditions on the interface are mass conservation, balance of stresses, and the Beavers-Joseph-Saffman condition. The splitting method involves single and decoupled Stokes and Biot solves at each time step. The subdomain problems use Robin boundary conditions on the interface, which are obtained from the transmission conditions. The Robin data is represented by an auxiliary interface variable. We prove that the method is unconditionally stable and establish that the time discretization error is $\mathcal{O}(\sqrt{T}\Delta t)$, where $T$ is the final time and $\Delta t$ is the time step. We further study the iterative version of the algorithm, which involves an iteration between the Stokes and Biot sub-problems at each time step. We prove that the iteration converges to a monolithic scheme with a Robin Lagrange multiplier used to impose the continuity of the velocity. Numerical experiments are presented to illustrate the theoretical results.