An extended finite element method for coupled Darcy–Stokes problems

Abstract

In this article, we propose an extended finite element method to deal with coupled Darcy-Stokes problems with straight or curved interfaces. A mixed approximate form is presented based on Brezzi-Douglas-Marini finite element space and piecewise constant function space for Darcy equations, and a mixed approximate form is shown based on piecewise quadratic function space and piecewise constant function space for Stokes equations. By introducing some stabilization terms, we prove that the inf-sup condition for the discrete coupled problem is valid. Moreover, the optimal convergence is derived. Finally, some numerical examples are presented to demonstrate our theoretical results.