Analysis of the diffuse interface method for the Stokes-Darcy coupled problem

We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy's law in the primal formulation. To solve this problem numerically, we use the diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width $\epsilon$ around the Stokes-Darcy interface. We prove the convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive the rates of the convergence. This is performed by analyzing the modeling error of the diffuse interface approach at the continuous level, and by deriving the a priori error estimates for the diffuse interface method at the discrete level. The convergence rates are also derived computationally in a numerical example.