Three parameter-robust virtual element numerical schemes for a Stokes–Biot fluid-poroelastic structure interaction model

¹ Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, 710129 Xi’an, P.R. China
² MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, 710129 Xi’an, P.R. China

^* Corresponding author: zhaojf@nwpu.edu.cn

Received: 17 June 2024
Accepted: 23 June 2025

Abstract

In this work, we develop and analyze three virtual element schemes of the fluid-poroelastic structure interaction model, in which the free fluid is described by the Stokes equations based on the velocity-pressure formulation and the flow in the poroelastic material is modeled by the Biot equations based on the displacement-pressure, displacement-pressure-flux, displacement-pressure-total pressure three formulations, respectively. By adopting a projection-based parameter-free stabilization (to circumvent the discrete inf-sup condition), introducing a Lagrange multiplier (to impose weakly the continuity of normal fluxes), and incorporating the total pressure as an unknown in the poroelastic region, we not only give three virtual element semi and fully discrete schemes and their well-posedness, respectively, but also derive a priori error estimates to demonstrate that our proposed schemes are free of a small viscosity, Poisson locking (as λ → ∞) and pressure oscillation (in the case of c₀ = 0 and a low permeability). Moreover, these schemes also support general polygonal meshes and arbitrary space approximation orders. Numerical experiments under both matching and non-matching meshes are presented to validate the performance of these schemes.