A Stable Nonconforming Mixed Element Method For Coupled Navier-Stokes And Darcy Flows

The Navier-Stokes equation and Darcy’s law are two important equations in describing fluid motion, they can describe the movement of viscous fluid in a given area and within the boundaries.They have very important application in many fields, such as the exploit of oil, natural gas and delivery of the groundwater. In recent years,the research on coupled Navier-Stokes and Darcy flow problems become a hot issue not only in theoretical studies but also in numerical analysis, and there are many articles on this.In this article,we mainly study the mathematical and numerical solution of coupled Navier-Stokes and Darcy flows.The model of this problem is based on imposing the Navier-Stokes equations in the fluid region and Darcy’s law in the porous media region, coupled with the appropriate interface conditions, which consists of the Beavers-Joseph-Saffman condition,the continuity of flux and the balance of forces.For the numerical solution of the this coupling prob-lem, there are still many difficulties so far,for example, the finite element space of speed and pressure must satisfy the Babuska-Brezzi stable condition, as well as the interface processing, especially mass conservation conditions.Therefore, we select Crouzeix-Raviart mixed finite element method, it has the following advantages:It satisfies Ladyzhenskaya-Babuska-Brezzi Conditions when com-bined with the piecewise constant, and is conserved in the space of piecewise.This paper has five chapters.We first use the actual physical background of the issues build mathematical model,get the relevant weak formulation and gives the Existence and Uniqueness conditions.Then,use a unified manner to approximate the velocity in the entire domain by nonconforming Crouzeix-Raviart element,along with piecewise constant element for the pressure.By adding penalizing term of velocity and deriving the discrete LBB condition,we get the solution of discrete scheme and prove existence and uniqueness of finite element solution.Then,an optimal priori error estimate is presented.Finally,there are some numerical examples to verity the convergence property of this scheme.