Pressure-robust Mixed Finite Element Method For Darcy-Stokes Coupling Problem

In this paper,after proveing that the Stokes Darcy coupling problem has the property of pressure robust,a pressure robust mixed element scheme is proposed undre the triangle mesh.the new scheme statified that the velocity error is inde-pendent of pressure in the sense of discrete norm.We select the nonconforming crouzeix raviart element in Stokes region and the lowest order raviart Thomas el-ement in Darcy region as discrete space.The two region are connected by beavers Joseph Saffman conditions,which is implied by bilinear form.In recent years,most of the mixed element methods have relaxed the diver-gence constraint and treated the continuous divergence free weakly as discrete divergence free.Therefore,the consistent error of pressure is introduced in the velocity error estimation of these mixed element methods.the pressure robust property is not considered in these methods.The main idea is to use the velocity reconstruction operator to ensure the divergence free in the discretization,and modify the body force term at the right hand.The modification does not change the stiffness matrix at the left end,but has the same convergence order as the speed and pressure of the traditional mixed element.Finally,some extreme numerical examples with small viscosity coefficients are used to verify the numerical convergence order of the analysis,and the superiority of the pressure robust scheme is illustrated.