Researches On Finite Element Methods For Two Types Of Nonlinear Free Flow And Porous Flow Coupling Model

In recent years,it is becoming more and more concerned in building suitable mathematical and numerical models for the fluid movement,which flows back and forth across a free fluid region and a porous medium.The models have a wide arrange of applications such as hydrology[14],environment science[26]and biofluid dynamics[32].Physically,the models consist of a coupled problem with two physical systems interacting across some interfaces.The simplest mathemat-ical formulation for the coupled problem is the coupled Stokes and Darcy flows problem,which imposes the Stokes equations in the fluid region and the Darcy's law in the porous media region,coupled with appropriate interface conditions.The conditions consist of mass conservation,the balance of normal forces and the Beavers-Joseph condition[6].Furthermore,in[72],Saffman proposed the Beavers-Joseph-Saffman conditions which are more suitable to the mathematical theory than the Beavers-Joseph conditions.There are many numerical methods for the coupled Stokes and Darcy flow problems which have been studied deeply.For example,the classical finite el-ement methods[3,54,77],the non-conforming finite element methods[68],the domain decomposition methods[7,13,19,26,78],the Lagrange multiplier and mixed element methods[12,42,44,55,61],the mixed finite element method combining with the discontinuous Galerkin method[66,67],the discontinuous Galerkin method combining with mimetic finite difference method[58],the spec-tral method[80],the pseudospectral least squares method[52],and many other numerical methods[17,20,25,31,43,45,59,63,74].All these papers only con-sider the linear model with single component flow,and just velocity and pressure appear in the model.However,in many cases,the simple linear coupled Stokes and Darcy flow model can not reasonably describe the objective physical phenomena.Therefore,it is necessary to further study the more complex model for coupling of free flow and porous media flow.We know that,in the porous media region,the Darcy's law describes the linear relationship between the velocity of the fluid and the gra-dient of pressure.The relationship is valid under the assumption that fluid flows are very slow,and the inertial terms(nonlinear terms)can be ignored.In 1901,by flow experiments in sandpacks observed in the bag,Forchheimer[36]found that for moderate Reynolds(Re(?)1 approximately),there will be a nonlinear relationship between pressure gradient and velocity,when Darcy's law is not adequate.For this case,he propose Darcy-Forchheimer equation ?K-1u+??|u|u + ?p = f to describe the nonlinear quadratic relationship of fluid velocity and pressure gra-dient in high velocity flow[4],where u is the fluid velocity,?p is the pressure gradient,?>0 is the fluid viscosity,K is the permeability tensor,?>0 is a dynamic viscosity,p>0 is the density of the fluid,f is a term related to body forces.This equation is widely used in the simulation of the vicinity of gas wells[2].There is only one paper[1]considers the coupling problem when the porous media flow region satisfies the Darcy-Forchheimer's law.In this article,Amirat proposed a mixed formation to the coupled comm-pressible Navier-Stokes and Darcy-Forchheimer equations with heat transfer and proved the well-posedness of the method,but he didn't give the error estimate.In this thesis,we study stabilized mixed finite element method for coupled Stokes and Darcy-Forchheimer flows,it follows the idea of same finite element,use P1 nonconforming Crouzeix-Raviart functions for each element to approximate the velocity and constant functions for each element to approximate the pressure.The stability of the format can be maintained by adding a penalty only related to the boundary of the unit.The reason why we use the nonconforming Crouzeix-Raviart element to solve the model is that it can be used to solve both Stokes problem[24]and Darcy-Forchheimer peoblem[81],then we can use matched grid-s across the interface and the same finite element spaces in both regions,and the interface conditions can be satisfied easily.The discontinuous finite elemen-t methods[16,47,66]have the similar properties,but more degrees of freedom are needed.What's more,from[81]we know that the Crouzeix-Raviart elemen-t method have many good convergence properties which is suitable for solving Darcy-Forchheimer model.With regard to the numerical analysis of the Darcy-Forchheimer model,Gi-rault and Wheeler[49]have proved the nonlinear operator(?)=?/?K-1v+?/?|v|v is monotone,coercive and hemi-continuous,and proved the existence and unique-ness of the solution of Darcy-Forchheimer model.We extend the definition of the operator to the whole coupled problem,so as to get the existence and uniqueness of the solution of the coupled Stokes and Darcy-Forchheimer flow problem.As for the error estimation of the Darcy-Forchheimer model,the absolute value esti-mations of the monotone operator is mentioned in[50,64,73].These estimations are meanly used to analysis the properties of the nonlinear Forchheimer term,which is f(v)= |v|v.We use these estimations for the overall coupling problem error analysis,and get the optimal order error estimates.For details see Chapter 1.At present,the problem of groundwater pollution caused by leaky under-ground storage tanks,chemical spills and many human activities has been paid more and more attention.This requires mass transfer equations to the coupled Stokes flow and Darcy flow model to describe the phenomenon.The coupled Stokes and Darcy flows with transfer model can also be used to describe the mis-cible displacement of one fluid by another in a vuggy porouous medium,and a variety of industrial processes associated with filtration.The coupled problem consist of two coupling meanings:the coupling of two domains and the coupling of flow and transport.Therefore,the fully coupled system becomes nonlinear and complex.For the coupled Stokes and Darcy flows with transfer model,there are only two articles[16,79]have studied its numerical method,but the viscosity of the fluid was assumed to be independent of concentration,this assumption decouples the flow equations from the concentration equation.The assumption reduces the difficulty of analysis and calculation,but it does not conform to the actual physical phenomenon.In this thesis,the viscosity of fluid in the coupled Stokes and Darcy flows with transfer model depends on the solute concentration,which greatly increases the nonlinearity of the problem.We use two methods to deal with the model:In Chapter 2,we use the idea of same finite element using the nonconforming piecewise Crouzeix-Raviart finite element,piecewise constant and conforming piecewise linear finite element to approximate velocity,pressure and concentration respectively.In Chapter 3,we propose a partitioned method with different subdomain time steps for differential equations in the system.The two methods have different advantages,the method using the idea of same finite element use less variables and requires less degree of freedom and by using a new formulation d(u;·,·),the lowest order format can be used to calculate.While the partitioned method with different subdomain time steps allows us to decouple the whole system into 3 small systems,which reduces the size of the problem and it is suitable for parallel computing.The two methods have considerable difficulty in both stability analysis and error estimation,for the partitioned method with different subdomain time steps,the error analysis also provides a guidance on the ratio of the time step sizes with respect to the ratio of the phiysical parameters,which gives a good reason for the selection of ratios of the time step sizes.For details see Chapter 2 and Chapter 3.The partitioned method with different subdomain time steps is an effective method for solving the coupled free fluid region and porous medium model.This approach is an optimization of the different regional partitioning methods pro-posed by Mu and Zhu[62]in 2010.In their article,Mu and Zhu assume that the time step sizes are all the same,and decouple the free flow region and the porous medium region,and solve the equations in each region.In[75],Shan etc.use different time steps in different subdomains following the fact that in most cases the changes in the physical quantities in the flow field of the porous media are much slower than the change in the physical quantity in the free flow region.That means using larger time step in the porous media region and shorter time step in the free flow region.This method is also used by Rybak and Magiera in[70],and the effectiveness of the method is verified by some practical examples.However,in these two articles,there is no clear explanation of the way to take the rate of different time steps,which brings limitations to the method.We will solve the problem in §3.4.On the other hand,higher order partitioned methods and many application-specific partitioned methods have been widely concerned in[13,20].The outline of the thesis is as follows.In Chapter 1,we present a stabilized mixed finite element method for solving the coupled Stokes and Darcy-Forchheimer flows problem.The approach utilizes the same nonconforming Crouzeix-Raviart mixed element for the velocity and piecewise constant for the pressure on the entire domain,and add a penalty term to guarantee the stability.We drive a discrete inf-sup condition and establish the existence and uniqueness of the problem.Optimal-order a prior error estimates are obtained based on the monotonicity owned by Forchheimer term.Finally,numerical examples are given to verify the the theoretical analysis,and verify the convergence of the iterative algorithm,and get the solution that the parameter?,? and the same permeability matrix K have no influence on the stability of the solution.In Chapter 2,we apply the stabilized mixed finite element method proposed in Chapter 1 to the coupled Stokes and Darcy flows with transfer problem.The approach utilizes the same nonconforming Crouzeix-Raviart mixed element for the velocity,piecewise constant for the pressure and conforming piecewise linear finite element for concentration on the entire domain,and add the penalty term proposed in Chapter 1 to guarantee the stability.The existence,uniqueness of the approximate solution are obtained.No assumption on the boundness of the infinity norms of approximate velocity or concentration or the restriction about the time step and spatial meshsize is needed due to a new bilinear form d(u;·,·),and we get optimal order a prior error estimates for the velocity and concentration.Finally,numerical examples are used to verify the theoretical analysis,and a practical example is used to reflect the effectiveness of our method.In Chapter 3,we present a decoupled finite element algorithm with different time steps on different physical variables for the model proposed in Chapter 2.The numerical algorithm decouple on the interface and calculate the partial dif-ferential equations in each part with different time steps.A careful error analysis provides a guidance on the ratio of the step sizes with respect to the ratio of physical parameters.Under a modest time step restriction in relation to physical parameters,stability of the method is obtained,and optimal order a priori error estimates are derived.Numerical examples are presented to verify the theoret-ical results and illustrate the effectiveness of the decoupled algorithm of using different time steps.In Chapter 4,we summarize the work done in this thesis,and briefly intro-duce the current research on the coupling problems of free flow and porous media,and establish our future research directions.