Quadratic discontinuous finite volume element schemes for Stokes-Darcy problems

Abstract

In this paper, we design three quadratic discontinuous finite volume element algorithms for the Stokes-Darcy problem. The key idea of the algorithms is to take the discontinuous function as trial function in the finite volume element method, and then to combine it with the discontinuous Galerkin method with three different types of internal penalties (incomplete, nonsymmetric, and symmetric), respectively. With the help of special mappings, we built up a bridge between the bilinear form of DFVM and that of discontinuous Galerkin method, which simplifies the analysis and obtains the well-posedness of the discrete DFVM problems. Then, we strictly demonstrate that both the broken $H^1$ norm errors of velocity and piezometric head and the standard $L^2$ norm error of pressure of the presented scheme converge to the optimal order. Finally, a series of numerical experiments are carried out to validate the results of the theoretical analysis, and to verify that the proposed scheme exhibits the characteristic of mass conservation, as well as the effectiveness of using non-matching mesh for calculations on the common interface for coupled flow problems.