A Crank–Nicolson discontinuous finite volume element method for a coupled non-stationary Stokes–Darcy problem

Abstract

In this paper, we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a Crank–Nicolson discretization in time, and we obtain the solution at the first time step by applying a first order backward Euler method to solve the coupling of time-dependent Stokes–Darcy problem. The proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by the piecewise constant elements. Then, the unique solvability of the approximate solution for the discrete problem is derived. In addition, the optimal error estimates of the full discretization in standard L2− norm and broken H1− norm are obtained. Finally, a series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, adaptive mesh refinement, mass conservation, capability to conveniently deal with complicated geometries with different types of boundary conditions, and the applicability to the problems with realistic parameters. In particular, the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain, it encompasses both fractures with high permeability which act as the prior flow conduit, and fractures with low permeability as a flow barrier.