A stabilized finite volume element method for a coupled Stokes–Darcy problem

Abstract

In this paper, we present a stabilized finite volume element method with the conforming finite element triples P1–P0–P1 and P1–P1–P1 for approximating the velocity, pressure, and hydraulic head of a coupled Stokes–Darcy problem. The proposed method is convenient to implement, computationally efficient, mass conserving, optimally accurate, and able to handle complex geometries; therefore, this method has great potential to be useful for realistic problems involving coupled free flow and porous media flow. To offset the lack of the inf-sup condition of the P1–P0 and P1–P1 elements for the Stokes equation, a parameter free stabilization term is added to the discrete formulation. Stability and optimal error estimates are proved based on a bridge built up between the finite volume element method and the finite element method. An element level implementation of the stabilization term is discussed so that an existing code package can be conveniently modified to handle the stabilization procedures. A series of numerical experiments are provided to illustrate the above features of the proposed method, the theoretical results, and the realistic applications.