A divergence free weak virtual element method for the Stokes–Darcy problem on general meshes

Abstract This paper presents a weak virtual element method on general meshes for the Stokes–Darcy problem with the Beavers–Joseph–Saffman interface condition. The velocity is discretized by the H ( div ) virtual element. The pressure is approximated by discontinuous piecewise polynomials. Besides, a polynomial space on the element faces is introduced to approximate the tangential trace of the velocity in the Stokes equations. The velocity on the discrete level is exactly divergence free and thus the exact mass conservation is preserved in the discretization. The well-posedness of the discrete problem is proved and an a priori error estimate is derived that implies the error for the velocity in a suitable norm does not depend on the pressure. A series of numerical experiments are reported to illustrate the performance of the method.