Analysis and finite element approximation of a diffuse interface approach to the Stokes--Biot coupling

TL;DR

This paper introduces a diffuse interface approach for modeling the interaction between poroelastic structures and free fluid flow, using finite element methods to ensure convergence and applicability to complex biological systems.

Contribution

The paper presents a novel diffuse interface model for Stokes--Biot coupling and proves its convergence to both the continuous diffuse and sharp interface models.

Findings

Finite element discretisation converges to the diffuse model.

Diffuse model converges to the sharp interface model.

Numerical examples confirm error estimates and demonstrate biological application.

Abstract

We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $\mathcal{O}(\varepsilon)$ around the interface; this allows the equations to be posed uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretisation of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We in turn prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.