Numerical Analysis Of Modular Grad-div Stabilization Methods For The Navier-stokes/Darcy Coupling Problems

In recent years,a lot of attention has been attracted to the coupling of incompress-ible flow and porous medium flows.This kind of physical phenomenon widely exists in the actual industrial engineering,for example,it can simulate the pollution of ground-water by pollutants in rivers,and it can also simulate the penetration of blood flow between blood vessels and organs.Therefore,it is significant to propose appropriate numerical methods for the coupling problem of free flow and porous media flow.In mathematics,Navier-Stokes/Darcy models are usually used to describe these phenom-ena.At present,many researchers have proposed a large number of numerical methods for the coupling problem of free flow and porous media flows.The Navier-Stokes/Darcy coupling problem consists of non-stationary Navier-Stokes equation,Darcy's law and suitable interface conditions.The conservation of mass condition,balance of forces,and Beavers-Joseph-Saffman(BJS)condition are imposed at the interface.However,there are many problems in the numerical solution of the coupling of the two equations.First,the Navier-Stokes equation is nonlinear,so we need to deal with the nonlinear term by iterative method.Secondly,there is the coupling problem of velocity and pressure in solving the Navier-Stokes equation.Thirdly,there are variables from the two simulation equations at the interface of the two fluids,which is also difficult to solve numerically.In this paper,the non-stationary Navier-Stokes/Darcy coupling problem is studied as follows.In this paper,first-order and second-order(BDF2)modular grad-div stability meth-ods are proposed for Navier-Stokes/Darcy and Stokes/Darcy coupled models,respec-tively.The key idea of this method is to add two zero stable terms to the continuity equation.In the fully discrete finite element scheme,the discrete stability terms is not zero,which will weaken the influence of pressure on velocity error and improve the accuracy of numerical solution.This method is divided into two steps.In the first step,an intermediate velocity is introduced to solve the decoupled coupling equation.In the second step,a modular grad-div stabilization step is introduced.We prove the stability and convergence under a time step and space step restriction of the (?t)/h?C.The modular grad-div stabilization method not only retains the advantages of the s-tandard grad-div stabilization,but also avoids solver breakdown due to the increase of stability parameters.Finally,a numerical tests are set to verify the effectiveness of the theoretical analysis,and it is found that the modular grad-div stabilization method can improve the computational efficiency.