A unified mixed finite element approximations of the Stokes–Darcy coupled problem

Abstract

In this paper we develop and analyze a unified approximation of the velocity–pressure pair for the Stokes–Darcy coupled problem in a plane domain. It is well known that, stable finite element approximations for the Stokes problem may not be appropriate for Darcy problem and for the coupling of fluid flow (modeled by the Stokes equations) with porous media flow (modeled by the Darcy equation), and therefore, different spaces are commonly used for the discretizations of the Darcy and the Stokes problems. In this work we proposed a modification of the Darcy problem which allows us to apply the classical Mini-element to the whole coupled Stokes–Darcy problem. The proposed method is probably one of the cheapest method for continuous approximation of the coupled system, has optimal accuracy with respect to solution regularity, and has simple and straightforward implementations. Numerical experiments are also presented, which confirm the excellent stability and accuracy of our method.