An absolutely stable weak Galerkin finite element method for the Darcy–Stokes problem

Abstract

In this paper, we apply the weak Galerkin (WG) finite element method to the Darcy–Stokes equations. This method provides accurate approximations for the velocity and the pressure variables. General polygonal or polyhedral partitions can be applied in this method. The finite element space which is made up of piecewise polynomials is easy to be constructed. These advantages make the weak Galerkin finite element method efficient and highly flexible. Optimal rates of convergence for the velocity function u and the pressure function p are established in corresponding norms. In addition, the convergence rates are independent of the viscosity parameter ϵ. Several numerical experiments are provided to illustrate the robustness, flexibility and validity of the weak Galerkin finite element method.