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.

翻译：我们研究以原始形式Biot模型描述的多孔弹性结构与用时变不可压缩Stokes方程描述的自由流动流体之间的相互作用。我们提出一种扩散界面模型，其中通过相场函数将耦合问题弱形式中的每个积分写在整个包含Stokes区域和Biot区域的域上。该相场函数在界面周围宽度为$\mathcal{O}(\varepsilon)$的扩散区域内从1连续过渡到0；这使得方程能够在整个域上统一表述，并避免了跟踪子域或它们之间的界面。我们证明了扩散界面模型有限元离散化在加权范数下向连续扩散模型的收敛性；此处权重是到扩散界面距离的幂次函数。我们进一步证明了连续扩散模型向标准锐界面模型的收敛性。数值算例验证了所证明的误差估计，并展示了该方法在通过复杂网络的流体流动（描述Willis环中的血液循环）中的应用。