A stabilized finite element method for the Navier--Stokes/Darcy coupled problem

Abstract

In this work, we propose, analyze, and numerically verify a new stabilized finite element method for the Navier--Stokes/Darcy coupled problem, which models a fluid flowing through a free medium into a porous medium. At the interface between both domains, we impose mass conservation, the balance of normal forces, and the well-known Beavers--Joseph--Saffman con- ditions [9]. The stabilization terms are defined using Galerkin's least-squares stabilization for the Navier--Stokes equation and Masud--Hughes stabilization [30] for the Darcy equation. This new discrete scheme employs equal-order elements to approximate the velocity and pressure of the fluid and generalizes the scheme recently analyzed for the linear case in [4]. The well-posedness of the discrete problem is established via fixed-point theorems under small data conditions, and we prove optimal error estimates in natural norms. Finally, we present numerical examples to confirm the expected theoretical convergence orders in cases where a manufactured solution is available, as well as to demonstrate the effectiveness of our scheme in a physical model with varying permeability fields.