Coupling Arbogast–Correa and Bernardi–Raugel elements to resolve coupled Stokes–Darcy flow problems

Abstract

This paper presents a finite element method for solving coupled Stokes–Darcy flow problems by combining the classical Bernardi–Raugel finite elements and the recently developed Arbogast–Correa (AC) spaces on quadrilateral meshes. The novel weak Galerkin methodology is employed for discretization of the Darcy equation. Specifically, piecewise constant approximants separately defined in element interiors and on edges are utilized to approximate the Darcy pressure. The discrete weak gradients of these shape functions and the numerical Darcy velocity are established in the lowest order AC space. The Bernardi–Raugel elements (BR1,Q0) are used to discretize the Stokes equations. These two types of discretizations are combined at an interface, where kinematic, normal stress, and the Beavers–Joseph–Saffman (BJS) conditions are applied. Rigorous error analysis along with numerical experiments demonstrate that the method is stable and has optimal-order accuracy.