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

TL;DR

This work addresses FPSI by developing a Robin-Robin domain-decomposition approach that blends Stokes fluid dynamics with Biot poroelasticity, enforcing interface continuity via Robin transmission conditions and an auxiliary data variable. The non-iterative scheme is proven unconditionally stable and to achieve a time-discretization error of ${\mathcal O}(\sqrt{T}\,Δt)$ in the quasistatic setting, while an iterative variant converges to a monolithic coupled formulation with a Robin Lagrange multiplier enforcing velocity continuity. The paper also derives a fully coupled monolithic scheme and proves its well-posedness, alongside numerical experiments that confirm first-order temporal accuracy and the robustness of the methods across a range of Robin parameters. These results advance stable, modular FPSI solvers capable of handling added-mass effects and poroelastic locking, and they extend insights from Stokes–Darcy to Stokes–Biot couplings with Beavers–Joseph–Saffman interfaces.

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 $\mathcal{O}(\sqrt{T}Δt)$, where $T$ is the final time and $Δt$ 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.

Figures (10)

• Figure 1: Schematic representation of a 2D computational domain.
• Figure 2: Example 1, left: computational domain and mesh; right: analytical solution.
• Figure 3: Example 1, convergence plots for fluid velocity ${\bf u}_f$ (left), Darcy velocity ${\bf u}_p$ (center), and displacement $\boldsymbol{\eta}_p$ (right) for $\gamma=1$ (first row), $\gamma=0.001$ (second row), and $\gamma=100$ (third row) computed by the non-iterative and iterative Robin-Robin methods and the monolithic method.
• Figure 4: Example 1, convergence plots for fluid velocity ${\bf u}_f$ (left), Darcy velocity ${\bf u}_p$ (center) and displacement $\boldsymbol{\eta}_p$ (right) for the non-iterative Robin-Robin algorithm for different values of $\gamma$.
• Figure 5: Example 1, convergence plots for fluid velocity ${\bf u}_f$ (left), Darcy velocity ${\bf u}_p$ (center) and displacement $\boldsymbol{\eta}_p$ (right) for the iterative Robin-Robin algorithm with 10 iterations per time step for different values of $\gamma$.

Theorems & Definitions (15)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Remark 4.1
• Theorem 5.1
• proof