Analysis and virtual element discretisation of a Stokes/Biot--Kirchhoff bulk--surface model

TL;DR

This paper develops and analyzes a stable virtual element method for a coupled 3D-2D Stokes/Biot-Kirchhoff model, proving its well-posedness and optimal convergence, with applications in immune isolation simulations.

Contribution

It introduces a novel virtual element discretization for a coupled bulk-surface model, establishing stability and convergence results.

Findings

Proved unique solvability of the continuous model.

Established stability and convergence of the discrete scheme.

Validated the method through numerical experiments and an application in immune isolation.

Abstract

We analyse a coupled 3D-2D model with a free fluid governed by Stokes flow in the bulk and a poroelastic plate described by the Biot-Kirchhoff equations on the surface. Assuming the form of a double perturbed saddle-point problem, the unique solvability of the continuous formulation is proved using Fredholm's theory for compact operators and the Babuska--Brezzi approach for saddle-point problems with penalty. We propose a stable virtual element method, establishing a discrete inf-sup condition under a small mesh assumption through a Fortin interpolant that requires only $H^1$-regularity for the Stokes problem. We show the well-posedness of the monolithic discrete formulation and introduce an equivalent fixed-point approach employed at the implementation level. The optimal convergence of the method in the energy norm is proved theoretically and is also confirmed numerically via computational experiments. We demonstrate an application of the model and the proposed scheme in the simulation of immune isolation using encapsulation with silicon nanopore membranes.