| There is an increasing interest in studing coupling incompressible fluid flow and porous media flow. This complex phenomena can be found in many sciences such as geosciences and health sciences. For example, the problems of inter-action of rivers with groundwater and the model of blood flow and organs has been researched in recent years. The fluid flow and the porous media flow are modeled by the Navier-Stokes equations and Darcy’s law under some assump-tions. However, The coupled models lead to various mathematical and numerical difficulties. First of all, there are many variables on the interface which lead to difficulties of modeling and solving this problem. Secondly, the local models may have complex or even non-linear forms. Thus, we should some iterative method to this model. Additionally, the Navier-Stokes and Darcy solutions have very different regularity properties, and the tangential velocity may be discontinous on the interface between the two regions. Furthermore, because the dimension of the interface is one less than the dimension of the domain, it is difficult to guarantee no loss in the regularity and the error orders when handling the weak formulation and the numerical scheme.In this paper, we use the normal component of velocity, the Beavers-Joseph-Saffman [43-45] law and balance of normal forces across the interface. In90’s of20th century, the early studies of the coupled problem have developed some numerical techniques [41,72], and both articles use Beavers-Joseph condition on interface. Beavers-Joseph condition is the law on the tangential fluid velocity across the interface between fluid and porous media, which is the best agree-ment with experiment evidence worked by Beavers and Joseph [44]. Before that, no-slippage along the interface is commonly used for the free fluid. Jones [37], ex-tended this to multidimensional flows. Saffman [43] proved this law theoretically and proposed the Beavers-Joseph-Saffman interface conditon.The coupled problem actually has two coupling meanings. One is the cou-pling of two domains. Different domains have different flow equations, different diffusion coefficients, and different sourses on sinks, and two domains interact each other only on the interface. The other one is the coupling of flow and transport. Therefore, the fully coupled system is very complicate. The cou-pling of flow and transport in a single domain has been extensively studied in [39,40,49,50,75-77]. In this paper, we mainly discuss the problem of the coupling of two domains.In general, coupled models can be solved in two types of approaches. One is direct method, and the other is to decouple the coupled models and apply appropriate local solver respectively. In past decades, many direct solvers have been proposed. Chidyagwai and Riviere apply continuous Galerkin (CG) and discontinuous Galerkin(DG) methods for the Navier-Stokes and Darcy models individually [6]. In [1], Girawlt and Riviere studied the approach by applying DG methods for both Navier-Stokes and Darcy models and gave the error estimate. In [73], a scheme called Crank-Nicolson/DG is analyzed for the time-dependent coupled problem. Moreover, in order to decrease the cost of computation for the coupled models, various decoupling techniques have been proposed. For instance. Quarteroni and Valli have extensively investigated heterogeneous domain decom-position methods for various coupled models [13]. The Lagrange multiplier [74] and the interface relaxation [38] approaches are widely used for decoupling multi-model problems.In recent years, two-grid methods are becoming popular in solving decoupled problems. The most important work among them is contributed by Mu and Xu [8] who proposed a two-gird decoupling method for Stokes/Darcy coupled models and gave the H1-error estimate. Furthermore, their method retains the same order of accuracy as solving the system directly when h=H3/2. Then, Cai, Mu and Xu continued to study the Navier-Stokes/Darcy coupled models in [9] and gave the same H1-error estimate as in [8].This thesis is about the steady coupled Navier-Stokes/Darcy model. The main contributing including:1. We construct a two-grid method for Navier-Stokes/Darcy model based on domain decoupling. There are many mathematical and numerical difficul- ties when we aim to solve the numerical solution of Navier-Stokes/Darcy model in practice. First of all, after the finite element discretization, the system is non-linear. Thus, we should apply some iterative methods to the system. However, in order to obtain high order accuracy, the number of iterations could be very large. Secondly, in order to increase the accuracy, refining the mesh is necessary, but the size of the system will be also increas-ing which leads to more difficulties. In this paper, We construct a two-grid method. Firstly, under first-order approximation, both theoretical anal-ysis and numerical experiments confirm that the the solution of two-grid method is of the same accuracy O(h) as the solution of coupled discrete problem for approximating the coupled Navier-Stokes/Darcy model with h=H2. Furthermore, under second-order approximation, theoretical anal-ysis confirms that the solution of two-grid method is of the same accuracy O(h2) as the solution of coupled discrete problem for approximating the coupled Navier-Stokes/Darcy model with h2=H3. Moreover, numerical experiments show that we can increase the match of coarse mesh and fine mesh size to h=H2.2. We construct a projection method for uH from coarse mesh to fine mesh. when we use the two-grid method proposed in this paper, integral on the interface have different basis functions which are defined on coarse and fine meshes respectively. Owing to the mismatch of the coarse and fine meshes, there are some difficulties in implementation. When the projection method is applied, the basis functions in the integral on the interface would be all defined on fine mesh which leads to convenient implementation. Further-more, this projection method can be also applied to multi-grid method.3. We construct a multi-grid method for Navier-Stokes/Darcy model. Based on the two grid method, this paper proposed a a multi-grid method and the theoretical analysis and numerical experiments is given. Firstly, under first-order approximation, both theoretical analysis and numerical experi-ments confirm that the solution of two-grid method is of the same accuracy O(hj) as the solution of coupled discrete problem for approximating the coupled Navier-Stokes/Darcy model with hj=hj2-1Furthermore, under second-order approximation, theoretical analysis confirms that the solution of two-grid method is of the same accuracy O(hj2) as the solution of cou-pled discrete problem for approximating the coupled Navier-Stokes/Darcy model with hj2=hj3-1. Moreover, numerical experiments show that we can increase the match of coarse mesh and fine mesh size to hj=hj2-1.This paper is organized as follows:The first chapter is the overview of the coupled Navier-Stokes/Darcy prob-lem. In this chapter, we state the background of the coupled Navier-Stokes/Darcy problem, the model of the steady coupled Navier-Stokes/Darcy problem, and fi-nite element method.In chapter2, we give the weak formulation and the finite element discretiza-tion of the the steady coupled Navier-Stokes/Darcy problem. The L2and Ⅱ1norm error estimates are given under first-order and second-order approximation. For instance, we apply MINI and Q2Q1finite elements to the discrete system,and give the basis functions on triangle and rectangle reference unit respectively. Fur-thermore,Guass integral formulation and stiffness matrices are also been given. In the end,numerical experiments verify the theoretical analysis.In chapter3,a two-grid method based on domain decomposition is proposed. Firstly, we give the method.Then,under first-order and second-order approxi-mation,we prove that the solution of the two-grid method is of the same accuracy O(h) and O(h2)as the solution of coupled discrete problem for approximating the coupled Navier-Stokes/Darcy model when the coarse and fine meshes satisfy h=H2and h2=H3respectively. Additionally, numerical experiments verify the theoretical analysis.In chapter4, a multi-grid method based on the two-grid method is given. Firstly, we give the method.Then,under first-order and second-order approxima-tion, we prove that the solution of the multi-grid method is of the same accuracy O(hj)and O(h2) as the solution of coupled discrete problem for approximat-ing the coupled Navier-Stokes/Darcy model when coarse and fine meshes satisfy hj=hj2-1and h2=hj3-1respectively. Additionally, numerical experiments verify the theoretical analysis.The last chapter is the conclusion and the future works. |
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