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

TL;DR

This paper introduces a diffuse interface method for the coupled Stokes-Darcy problem, proving its convergence to the sharp interface model and demonstrating its effectiveness through error estimates and numerical validation.

Contribution

The paper develops and analyzes a diffuse interface approach for Stokes-Darcy coupling, providing convergence proofs and error estimates, which are validated numerically.

Findings

Convergence of diffuse interface method to sharp interface model.

Derivation of a priori error estimates for discretizations.

Numerical validation confirming theoretical convergence rates.

Abstract

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 a 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 $\mathcal{O}(\epsilon)$ around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.