Some low order nonconforming mixed finite elements combined with Raviart–Thomas elements for a coupled Stokes–Darcy model

Abstract

In this paper, we discuss numerical methods to solve a coupled Stokes–Darcy system. The deformation tensor form of Stokes equations is used to describe the fluid flow motion. The mixed form of elliptic equation is applied to describe the porous media flow motion. We propose finite element methods for the coupled problem. For Stokes equations, one component of the velocity is approximated by Crouzeix–Raviart element or Rannacher–Turek element, and the other component is approximated by conforming P1 or Q1 element; pressure is approximated by piecewise constants. For the mixed form of elliptic equation, the lowest order triangular/quadrilateral Raviart-Thomas element is used. The discrete mesh is nonmatching on the interface. By Boland-Nicolaides trick, the inf-sup condition of the discrete problem is proved. Moreover, we construct a new interpolation operator to derive the a priori error estimate of the proposed finite element method. Numerical examples are also given to confirm the theoretical results.