| Fractures, which are secondary gap structures in karst medium, occupy the main part of the total porosity [76]. They are not only the occurrence of groundwater resources, but also the places where the problems of environ-mental pollution occur."The study of Karst aquifer in our country faces the opportunities and challenges " by Daoxian Yuan in [85] shows that the trends of water pollution are intensifying in karst aquifer system and con-tradictions between economic development and environmental pollution are also increasingly prominent. Improving the understanding of Karst aquifer flow system will help to control potential and inherent vulnerability factors of groundwater resources.Water flow in the fractures and the surrounding medium are closely linked, therefore, the coupled model is often used in current studies of ground-water flow in fractured media. The coupling term is the exchange flux occur-ring in the junctures between the fractures and the porous media[5,6,7,8.10,12,73]. In recent years, the coupled continuum pipe-flow/Darcy (CCPF) model has been presented to describe the flow in Karst aquifer system [14], in which the single-phase flow is assumed to meet Darcy’s law in porous media and the pipe-flow model is used to describe the fracture flow. The building process of CCPF model is given in the first chapter.Combining the idea of dual porosity, CCPF model has been solved by the Carbonate Aquifer Void Evolution (CAVE) code in [31]. CAVE solved the flow in the porous matrix by a finite difference scheme using MODFLOW [38] and the flow in conduit by a nonlinear finite difference discretization. It is demonstrated in [75] that the coupled (stationary) model is well-posed in two spatial dimension but ill-posed in three spatial dimension. With mathe-matical regularity of the problem, Cao et al.[14] have applied finite element approximation to the two-dimensional case and presented optimal conver-gence rates in the L2and H1norms. Since the finite volume element method inherits the physical conservation laws of the original problem locally, Chap-ter2presents finite volume element method to approximate CCPF model. Because of the existence of Dirac delta function, the analytic solution of CCPF may have anisotropic behavior near the pipe region, which means that the solution of Darcy model in the porous media varies significantly along the direction parallel to y-axis and is smooth along the direction parallel to x-axis, Chapter3gives Wilson nonconforming element on anisotropic grid combining with conforming finite element on regular grid for CCPF model. Due to the existence of Dirac delta function, the analytic solution of Darcy equation in CCPF model is smooth along the direction parallel to x axis but with low regularity near the pipe-flow region along the direction parallel to y axis. Considering the physical features of solution, Chapter4introduces a new nonconforming finite element to solve it.For solving nonlinear equations, two-grid method is a high efficient and high accurate algorithm. With the technique of this method, solving a non-linear problem on the fine grid is reduced to solving a linear system on the fine grid and a small nonlinear system on the coarse grid. For the nonlin-ear reaction-diffusion problem, Chapter5gives the two-grid expanded mixed finite element method. Chapter6presents the two-grid expanded mixed finite element method for the semilinear elliptic problem. For the Darcy-Forchheimier model, the two-grid method can not be used directly owe to the nonlinear term including a norm function|·|without the continuous derivatives, Chapter7modifies the two-grid method by adding a very small and positive parameter ε and uses the modified method to obtain the blocked- center finite difference solutions of Darcy-Forchheimier model.The outline of the dissertation is as follows.In Chapter1, some preliminaries are given and discussed. A mathemat-ical model called the coupled continuum pipe-flow/Darcy (CCPF) model is given to describe the flow in Karst aquifer, in which Darcy model is applied to study the flow in porous media and pipe-flow model is used to study the flow in conduit region. A mathematical algorithm called two-grid method is demonstrated in details and applied to solve nonlinear problems. Its idea is basically to use a coarse space to produce a rough approximation of the solution and then use it to obtain a linearized system on a fine grid.In Chapter2, the finite volume element method is used to solve CCPF model. Firstly, the finite volume element approximation scheme is given, then existence and uniqueness of the approximation solution are derived. Using integration formula properly, the optimal error estimates are obtained in certain discrete norms. Based on some decoupling techniques, the coupled system can be solved by obtaining the solutions of Darcy model and pipe-flow model separately. Finally some numerical experiments are presented to show the efficiency of the scheme. This work is published in "Numerical Methods for Partial Differential Equations"[50].In Chapter3, Wilson element on anisotropic mesh is applied to solve the Darcy equation of CCPF model. Near the pipe-line we use an anisotropic mesh with a small meshstep on the y-direction and a large meshstep on x-direction to meet the low regularity of the analytic solution on y-direction. The existence and uniqueness are obtained for the approximation solution. The optimal error estimates are established in L2and H1norms indepen-dent of the regularity condition on the mesh. Numerical examples show the efficiency of our scheme. With the same number of nodal-points the results using Wilson element on anisotropic mesh are better than the same element on regular mesh, and also better than Q1,1element on regular mesh.In Chapter4, using a new nonconforming element we give a coupled numerical scheme for CCPF model. The numerical scheme is a combination of the standard finite element method for pipe-flow equation with the new nonconforming element for the Darcy equation. The existence and unique-ness of the solution of approximation scheme are obtained on the regular mesh. Optimal order error estimates are established in L2and H1norms. Based on three different subdivisions, some experimental examples show the convergence rates are consistent with the theoretical analysis. Superconver-gence results appear in a few experimental examples. On the same mesh, the numerical results using the new nonconforming finite element method for the Darcy equation of CCPF model are much better than using Q1,1element or Wilson element.In Chapter5, instead of solving a large system of nonlinear equations directly with expanded mixed finite element approximation for the following nonlinear reaction-diffusion problem, we shall consider a two-grid method, where Ω is a bounded, convex domain with C2boundary, ν is the unit exterior normal to аΩ, and K is a symmetric positive definite tensor.The feature of the two-grid method is that it allows one to execute all the nonlinear iterations on a system associated with a coarse spatial grid. The method discussed here extends the discretization technique to the non-linear reaction-diffusion equations based on expanded mixed finite element spaces defined on two grids of different sizes. This procedure is basically to use the coarse grid of size H to produce a rough approximation of the solution and then use it as the initial guess on the fine grid of size h. Error estimates of two-grid method are derived which demonstrate that the error is O(△t+hk+1+H2k+2-d/2), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimate is useful for determining an appropriate H for the coarse grid prob-lems. According to some numerical examples, it is obvious that the two-grid algorithm saves much CPU time and is a viable computational approach. This work is published in "Acta Mathematical Applicatae Sinica, English Series"[51].In Chapter6, we shall use a two-grid expanded mixed finite element method for the following semi-linear elliptic model.In order to obtain asymptotically optimal approximation results, we need to choose appropriate relations between the coarse grid mesh size H and the fine grid mesh size h. Therefore, with RTN and BDM mixed ele-ment, convergence results in Lq norm and H-s norm are firstly derived for the expanded mixed element approximation scheme of model problem. Se-quently, on the basis of the convergence estimates, the errors of solutions obtained from the two-grid algorithm is deduced as the principle of deter-mining a proper mesh size H for the coarse grid system. It is clearly shown that H=O(h1//2) is chosen in a sense of L2norm. This means that solv-ing a nonlinear elliptic problem is not much more difficult than solving one linear problem, since the work for solving the nonlinear problem is relatively negligible. This work is published in "Computers and Mathematics with Applications "[52].In Chapter7, we shall use a two-grid for the following nonlinear Darcy-Forchheimier model based on block-centered finite difference method, with the compatibility condition where p denotes the pressure and u the velocity of the fluid. n is the unit exterior normal vector to the boundary of Q,|·|represents the Euclidean norm, and|u|2=u·u. ρ,μ,, and β are scalar functions which denote the density of the fluid, its viscosity, and its dynamic viscosity, respectively. βis also called as the Forchheimer number. K is the permeability tensor function. For simplicity we suppose that K=kI, where k is positive and I represents the unit matrix.f(x)∈L2(Q), a scalar function, is the source and sink term.▽h(x)∈(L2(Ω))2, a vector function, represents the gradient of the depth function h(x)∈H1(Ω).fN(x)∈L2(аΩ), a scalar function, denotes the Neumann boundary condition, or the flux through the boundary.The key of using the two-grid method lies in the differentiable properties of nonlinear term. However, the nonlinear term of Darcy-Forchheimier model contains a norm function|·|without the continuous derivatives. Then, we consider to use a technique by adding a very small and positive parameter ε to obtain a modified nonlinear term with twice continuously differentiable with bounded derivatives through second order. Moreover, it is proved that the modified nonlinear term and its derivatives are uniformly bounded about the parameter ε. By taking a proper ε, we can get the optimal order error estimates in the discrete L2norm for the approximation scheme of Darcy-Forchheimier model. On the basis of the convergence estimates, the errors of solutions obtained from the two-grid algorithm is deduced as the principle of determining a proper mesh size H for the coarse grid system. According to the errors estimates in this paper, it is clearly shown that H=O(h1/2) is chosen in order to obtain asymptotically optimal approximation results. Some computational results are used to confirm the algorithm’s utility. From the number of iterations point of view, the two-grid algorithm, without loss any accuracy, is more efficient. |
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