地下耦合流动模型求解和非线性问题二重网格方法
|
摘要
裂缝属于岩溶介质次生空隙结构,占总空隙的主要部分[76].它们不仅是地下水的赋存空间,也是环境污染现象的发生场所.袁道先院士在“新形势下我国岩溶研究面临的机遇和挑战”[85]中提出,岩溶地区水环境污染趋势正在加剧,经济发展和环境污染之间的矛盾也日益突出.提高对地下水流动系统的认识,有助于研究控制地下水资源固有脆弱性的潜在因素.
裂缝中的水流动与周围介质中的水流动有密切联系,当前研究中多用耦合模型来刻画裂缝介质地下水流动过程,其中耦合项是交界处的流量交换项[5,6,7,8,10,12,73].地下水实验证明流量交换项与岩溶介质和裂缝区域中的水头差成正比,比例系数与岩溶介质的传导系数和裂缝的几何形态等有关.岩溶介质属于多孔介质,往往假设其单相流动满足达西定律[60,79],对于裂缝中的流动,使用管流模型来刻画,因此形成了管流／达西(CCPF)耦合模型[14,44,75],模型建立过程详见第一章.
结合双重孔隙率方法,CCPF模型可以通过碳酸盐岩含水层孔隙演化(CAVE)代码求解[31].CAVE使用了MODFLOW[38]有限差分格式.稳态的CCPF模型在二维区域是适定的[75].CCPF模型的正则性已被证明,且得到了该模型有限元解的最优阶的L2和H1模误差估计[14].本文第二章至第四章将给出求解CCPF模型的一些数值方法.CCPF模型满足质量守恒定律,所以第二章考虑使用保持质量守恒的有限体积元法求解该耦合模型.裂缝与周围介质的交界边附近,存在各向异性特征,第三章将使用基于各向异性网格的Wilson元方法.鉴于该耦合模型的解具有沿着裂缝走向连续和垂直裂缝走向间断的特点,第四章将提出一种新型的非协调元方法求解CCPF模型.
地下水模拟和油藏数值模拟经常使用非线性模型,很多数值方法可以用来求解这类模型,其中比较著名的是二重网格方法.该方法的主要思想是：首先在粗网格上求解一个小型的非线性方程,然后利用这个粗网格上的数值解构造细网格上的线性方程,将这个线性方程的解作为非线性问题的数值解.因此二重网格法可以把一个在细网格上求解非线性方程的问题转化成一个粗网格上求解非线性方程和一个细网格上求解线性方程的问题.该方法的主要特点是将所有非线性迭代都放在粗网格空间执行.如果粗网格非常粗的话,在粗网格上的计算量就可以相对忽略.二重网格法的误差分析决定了粗网格步长和细网格步长之间的关系,根据误差估计可知二重网格算法既能不损失精度,又能提高计算效率.
本文将在第五章至第七章使用二重网格法求解一些非线性的模型.针对含有压力和速度非线性项的反应扩散方程,第五章给出了二重网格扩展混合元方法.第六章研究了半线性椭圆方程的二重网格扩展混合元解.Darcy-Forchheimier模型含有速度的非线性项,但是这个非线性项不可导,无法直接使用二重网格法,第七章将改进二重网格法并用其求解Darcy-Forchheimier模型的块中心有限差分解.
本文的组织结构如下：
第一章,给出了两部分的预备知识.第一部分介绍了一个数学模型——连续介质耦合的管流／达西(CCPF)模型.CCPF模型可以用来描述岩溶含水层单相流动问题,其中达西模型被用于研究多孔介质中的水流动过程,管流模型是用来研究裂缝区域的水流动过程.第二部分介绍了一个数学方法——二重网格法.二重网格方法经常用来求解非线性问题,该方法基于粗细两套网格,首先使用粗网格产生一个粗略的近似,然后用它作为初始猜测值在细网格上构造一个线性系统,最后求解此线性系统来得到非线性问题的解.
第二章,使用有限体积元法求解CCPF问题.首先给出了有限体积元逼近格式,然后证明了解的存在唯一性,得到了离散模意义下的最优阶误差估计.最后使用一些数值实验验证了方法的有效性和收敛性.
第三章,使用基于各向异性网格的Wilson元求解CCPF模型中达西方程,基于正则网格的一维有限元求解CCPF模型中的管流模型.在管道区域附近,因为解析解沿着y方向正则性低,所以采用各向异性网格剖分方式.证明了数值解的存在性和唯一性,推导了L2和H1模的最优阶误差估计.数值算例表明了该耦合数值方法的有效性.在网格剖分节点相同的情况下,基于各向异性网格的Wilson元的结果比正则剖分下的Wilson元或Q1,1元的结果好一些.
第四章,构造了一种新的非协调元,其基函数在参考单元中沿着一个方向连续而沿着另一个方向间断.使用新的非协调元-标准协调元耦合的数值方法求解CCPF模型.在正则的网格剖分下得到解的存在性和唯一性,推导了在L2和H1范数下的最优阶误差估计.数值算例中,基于三种不同的网格剖分方式研究了这种耦合数值方法的收敛速度,得到数值结果与理论分析一致的结论.甚至在一些例子中出现了超收敛现象.基于同一种剖分网格,采用新型的非协调有限元方法求解CCPF模型中的达西方程,比使用Q1,1元或Wilson元的结果要好得多.
第五章,使用二重网格法对非线性反应扩散方程的扩展混合元格式进行了求解,方程形式如下：其中Ω是具有C2边界aQ的d(d≥2)维区域,v是aQ的单位外法线向量,p表示未知的压力,s是可压系数,K是多孔岩石蓄水层的传导率.
二重网格算法可以将细网格空间求非线性方程组转化为在粗网格空间求一个非线性解和在细网格空间求一个线性解的过程.算法的误差为O(△t+hk+1+H2k+2-d/2),其中h,H分别为细网格步长和粗网格步长,k是逼近空间多项式次数,△t是时间步长.此误差估计式有助于粗网格步长的选取.
第六章,使用二重网格扩展混合元方法求解下面的半线性椭圆模型.基于RTN和BDM元,首先推出该模型的扩展混合元逼近格式,然后推导Lq和H-s模的误差估计,以此为基础研究压力和速度的最优阶误差估计,估计式子可以作为H的取值依据.在L2范数下,H=(O)(h1/2)是比较好的选择,可以得到渐近的最佳逼近.这意味着求解非线性椭圆问题并不比解决一个线性问题困难得多,因为用于解决非线性问题的工作量可以相对忽略.
第七章,基于块中心的有限差分格式使用二重网格法求解下列非线性Darcy-Forchheimier模型.其中相容条件为这里p代表压力,u代表流体的速度,n是Ω边界的单位外法线向量,|·|表示欧几里得范数,且|u|2=u·u. p,μ和β是标量函数,分别代表流体的密度和粘性系数,口也被称作Forchheimer数,K代表孔隙度的张量函数.f(x)∈L2(Ω)代表源汇项.▽h(x)∈(L2(Ω))2代表深度函数h(x)∈H1(Ω)的梯度.fN(x)∈L2(аΩ)表示Neumann边界条件或通过边界的流量.
使用二重网格方法的关键在于非线性项的二阶可导性.然而,Darcy-Forchheimier模型的非线性项含有的｜·｜导致导数不存在.因此考虑使用一种改进技巧：增加一个非常小的正参数ε来获得修正的非线性项,使其具有连续两次有界的导数.事实证明,改进后的非线性项和它的导数关于参数ε是一致有界的.通过采取适当的ε可以得到在离散L2范数下的最优阶误差估计.
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.
裂缝中的水流动与周围介质中的水流动有密切联系,当前研究中多用耦合模型来刻画裂缝介质地下水流动过程,其中耦合项是交界处的流量交换项[5,6,7,8,10,12,73].地下水实验证明流量交换项与岩溶介质和裂缝区域中的水头差成正比,比例系数与岩溶介质的传导系数和裂缝的几何形态等有关.岩溶介质属于多孔介质,往往假设其单相流动满足达西定律[60,79],对于裂缝中的流动,使用管流模型来刻画,因此形成了管流／达西(CCPF)耦合模型[14,44,75],模型建立过程详见第一章.
结合双重孔隙率方法,CCPF模型可以通过碳酸盐岩含水层孔隙演化(CAVE)代码求解[31].CAVE使用了MODFLOW[38]有限差分格式.稳态的CCPF模型在二维区域是适定的[75].CCPF模型的正则性已被证明,且得到了该模型有限元解的最优阶的L2和H1模误差估计[14].本文第二章至第四章将给出求解CCPF模型的一些数值方法.CCPF模型满足质量守恒定律,所以第二章考虑使用保持质量守恒的有限体积元法求解该耦合模型.裂缝与周围介质的交界边附近,存在各向异性特征,第三章将使用基于各向异性网格的Wilson元方法.鉴于该耦合模型的解具有沿着裂缝走向连续和垂直裂缝走向间断的特点,第四章将提出一种新型的非协调元方法求解CCPF模型.
地下水模拟和油藏数值模拟经常使用非线性模型,很多数值方法可以用来求解这类模型,其中比较著名的是二重网格方法.该方法的主要思想是：首先在粗网格上求解一个小型的非线性方程,然后利用这个粗网格上的数值解构造细网格上的线性方程,将这个线性方程的解作为非线性问题的数值解.因此二重网格法可以把一个在细网格上求解非线性方程的问题转化成一个粗网格上求解非线性方程和一个细网格上求解线性方程的问题.该方法的主要特点是将所有非线性迭代都放在粗网格空间执行.如果粗网格非常粗的话,在粗网格上的计算量就可以相对忽略.二重网格法的误差分析决定了粗网格步长和细网格步长之间的关系,根据误差估计可知二重网格算法既能不损失精度,又能提高计算效率.
本文将在第五章至第七章使用二重网格法求解一些非线性的模型.针对含有压力和速度非线性项的反应扩散方程,第五章给出了二重网格扩展混合元方法.第六章研究了半线性椭圆方程的二重网格扩展混合元解.Darcy-Forchheimier模型含有速度的非线性项,但是这个非线性项不可导,无法直接使用二重网格法,第七章将改进二重网格法并用其求解Darcy-Forchheimier模型的块中心有限差分解.
本文的组织结构如下：
第一章,给出了两部分的预备知识.第一部分介绍了一个数学模型——连续介质耦合的管流／达西(CCPF)模型.CCPF模型可以用来描述岩溶含水层单相流动问题,其中达西模型被用于研究多孔介质中的水流动过程,管流模型是用来研究裂缝区域的水流动过程.第二部分介绍了一个数学方法——二重网格法.二重网格方法经常用来求解非线性问题,该方法基于粗细两套网格,首先使用粗网格产生一个粗略的近似,然后用它作为初始猜测值在细网格上构造一个线性系统,最后求解此线性系统来得到非线性问题的解.
第二章,使用有限体积元法求解CCPF问题.首先给出了有限体积元逼近格式,然后证明了解的存在唯一性,得到了离散模意义下的最优阶误差估计.最后使用一些数值实验验证了方法的有效性和收敛性.
第三章,使用基于各向异性网格的Wilson元求解CCPF模型中达西方程,基于正则网格的一维有限元求解CCPF模型中的管流模型.在管道区域附近,因为解析解沿着y方向正则性低,所以采用各向异性网格剖分方式.证明了数值解的存在性和唯一性,推导了L2和H1模的最优阶误差估计.数值算例表明了该耦合数值方法的有效性.在网格剖分节点相同的情况下,基于各向异性网格的Wilson元的结果比正则剖分下的Wilson元或Q1,1元的结果好一些.
第四章,构造了一种新的非协调元,其基函数在参考单元中沿着一个方向连续而沿着另一个方向间断.使用新的非协调元-标准协调元耦合的数值方法求解CCPF模型.在正则的网格剖分下得到解的存在性和唯一性,推导了在L2和H1范数下的最优阶误差估计.数值算例中,基于三种不同的网格剖分方式研究了这种耦合数值方法的收敛速度,得到数值结果与理论分析一致的结论.甚至在一些例子中出现了超收敛现象.基于同一种剖分网格,采用新型的非协调有限元方法求解CCPF模型中的达西方程,比使用Q1,1元或Wilson元的结果要好得多.
第五章,使用二重网格法对非线性反应扩散方程的扩展混合元格式进行了求解,方程形式如下：其中Ω是具有C2边界aQ的d(d≥2)维区域,v是aQ的单位外法线向量,p表示未知的压力,s是可压系数,K是多孔岩石蓄水层的传导率.
二重网格算法可以将细网格空间求非线性方程组转化为在粗网格空间求一个非线性解和在细网格空间求一个线性解的过程.算法的误差为O(△t+hk+1+H2k+2-d/2),其中h,H分别为细网格步长和粗网格步长,k是逼近空间多项式次数,△t是时间步长.此误差估计式有助于粗网格步长的选取.
第六章,使用二重网格扩展混合元方法求解下面的半线性椭圆模型.基于RTN和BDM元,首先推出该模型的扩展混合元逼近格式,然后推导Lq和H-s模的误差估计,以此为基础研究压力和速度的最优阶误差估计,估计式子可以作为H的取值依据.在L2范数下,H=(O)(h1/2)是比较好的选择,可以得到渐近的最佳逼近.这意味着求解非线性椭圆问题并不比解决一个线性问题困难得多,因为用于解决非线性问题的工作量可以相对忽略.
第七章,基于块中心的有限差分格式使用二重网格法求解下列非线性Darcy-Forchheimier模型.其中相容条件为这里p代表压力,u代表流体的速度,n是Ω边界的单位外法线向量,|·|表示欧几里得范数,且|u|2=u·u. p,μ和β是标量函数,分别代表流体的密度和粘性系数,口也被称作Forchheimer数,K代表孔隙度的张量函数.f(x)∈L2(Ω)代表源汇项.▽h(x)∈(L2(Ω))2代表深度函数h(x)∈H1(Ω)的梯度.fN(x)∈L2(аΩ)表示Neumann边界条件或通过边界的流量.
使用二重网格方法的关键在于非线性项的二阶可导性.然而,Darcy-Forchheimier模型的非线性项含有的｜·｜导致导数不存在.因此考虑使用一种改进技巧：增加一个非常小的正参数ε来获得修正的非线性项,使其具有连续两次有界的导数.事实证明,改进后的非线性项和它的导数关于参数ε是一致有界的.通过采取适当的ε可以得到在离散L2范数下的最优阶误差估计.
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.
引文
[1]T. Apel and M. Dobrowolski. Anisotropic interpolation with applications to the finite element method. Computing,47(1992),277-293.
[2]T. Arbogast, M. F. Wheeler, and I. Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal,34(1997),828-852.
[3]K. Aziz and A. Settari. Petroleum Reservoir Simulation. Applied Science Publishers LTD, London,1979.
[4]R. E. Bank and D. J. Rose. Some error estimates for the box method. SIAM J. Numer. Anal.,24(1987),777-787.
[5]G. Barenblatt, I. Zheltov, and I. Kochina. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech., 24(1960),1286-1303.
[6]S. Bauer, R. Liedl, and M. Sauter. Modeling of karst aquifer genesis:influence of exchange flow. Water Resour. Res.,39(2003), SBH6.1-SBH6.12.
[7]S. Bauer, R. Liedl, and M. Sauter. Modelling of karst development consider-ing conduit-matrix exchange flow, Calibration and reliability in groundwater modelling:coping with uncertainty. IAHS Publ,265(2000),10-15.
[8]J. Bear and A. Verruijt. Modeling groundwater flow and pollution. D. Reidel Pub Co, Norwell, MA,1987.
[9]C. Bi and V. Ginting. Two-grid finite volume element for linear and nonlinear elliiptic problems. Numer. Math.,108(2007),177-198.
[10]S. Birk, R. Liedl, M. Sauter, and G. Teutsch. Hydraulic boundary conditions as a controlling factor in karst genesis. Water Resour. Res.,39(2003), SBH2.1-SBH2.14.
[11]F. Brezzi, J. Douglas Jr., M. Fortin, and L. Marini. Two families of mixed finite elements for second order elliptic problems. Numer. Math.,47(1985), 217-235.
[12]E. Bobok. Fluid mechanics for petroleum engineers. Elsevier Sci, New York, 1993.
[13]Z. Cai. On the finite volume element method. Numer. Math.,28(1991), 392-402.
[14]Y. Cao, M. Gunzburger, F. Hua, and X. Wang. Analysis and Finite Element Approximation of a Coupled, Continuum Pipe-Flow/Darcy Model for Flow in Porous Media with Embedded Conduits. Numer. Methods Partial Differential Eq.,27(2011),1242-1252.
[15]Z. Chen. The control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media,1(2006),689-706.
[16]Z. Chen. Expanded mixed element methods for linear second-order elliptic problems(Ⅰ). RAIRO Model. Math. Anal. Numer.,32(1998),479-499.
[17]Z. Chen. Expanded mixed element methods for quasilinear second-order el-liptic problems(Ⅱ). RAIRO Model. Math. Anal. Numer.,32(1998),501-520.
[18]Z. Chen. BDM mixed methods for a nonlinear elliptic problem. J. Comp. Appl. Math.,53(1994),207-223.
[19]L. Chen and Y. Chen. Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods. J Sci. Comput.,49(2011), 383-401.
[20]N. Chen, M. Gunzburger, B. Hu, X. Wang, and C. Woodruff. Calibrating the exchange coefficient in the modified coupled continuum pipe-flow model for flows in karst aquifers. J. Hydrol,414-415(2012),294-301.
[21]Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, in the Computational Science and Engineering Series, Vol. 2, SIAM, Philadelphia,2006.
[22]Y. Chen, Y. Huang, and D. Yu. A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int J. Numer. Meth. Eng.,57(2003),193-209.
[23]H. Chen and B. Li. Superconvergence analysis and error expansion for the Wilson nonconforming finite element. Numer. Math.,69(1994),125-140.
[24]C. Chen and W. Liu, Two-grid finite volume element methods for semilinear parabolic problems. Appl. Numer. Math.,60(2010),10-18.
[25]C. Chen and W. Liu. A two-grid method for finite volume element approxima-tions of second-order nonlinear hyperbolic equations, J. Comp. Appl. Math., 233(2010),2975-2984.
[26]Y. Chen, H. Liu, and S. Liu. Analysis of two-grid methods for reaction diffusion equations by expanded mixed finite element methods. Int J. Numer. Meth. Eng.,69(2007),408-422.
[27]S. Chen, D. Shi, and Y. Zhao. Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes. IMA Journal of Numerical Analysis, 24(2004),77-95.
[28]S. Chen, Y. Zhao, and D. Shi. Anisotropic interpolations with application to nonconforming elements. Applied Numerical Mathematics,49(2004),135-152.
[29]S. H. Chou, D. Y. Kwak, and Q. Li. LP error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differential Eq.,19(2003),463-486.
[30]S. H. Chou and Q. Li. Error estimates in L2, H1 and L∞ in covolume methods for elliptic and parabolic problems:A unified aproach. Math. Comp., 69(2000),103-120.
[31]T. Clemens, D. Huckinghaus, M. Sauter, R. Liedl, and G. Teutsch. A com-bined continuum and discrete network reactive transport model for the sim-ulation of karst development. IAHS Publ,237(1996),309-318.
[32]C. N. Dawson and M. F. Wheeler. Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. Contempt Maths,180(1994), 191-203.
[33]C. N. Dawson, M. F. Wheeler and C. S. Woodward. A two-grid finite dif-ference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal., 35(1998),435-452.
[34]R. E. Ewing, Z. Li, T. Lin, and Y. P. Lin. The immersed finite volume ele-ment methods for the elliptic interface problems. Math. Comput. Simulations, 50(1999),63-76.
[35]R. E. Ewing, T. Lin, and Y. P. Lin. On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal.,39(2002),1865-1888.
[36]P. FORCHHEIMER. Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing., 45(1901),1782-1788.
[37]V. Girault and M. F. Wheeler. Numerical discretization of a Darcy-Forchheimer model. Numer. Math.,110(2008),161-198.
[38]A. Harbaugh. The U.S. Geological survey modular groundwater model-the GroundWater Flow Process. U.S. Geological Survey Techniques and Methods 6-A16 (2005).
[39]Y. He and J. Li. Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg.,198(2009),1351-1359.
[40]Y. He and W. Sun. Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations. SIAM J.Numer. Anal.,45(2007),837-869.
[41]Y. He and W. Sun. Stabilized finite element methods based on Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comp.,76(2007),115-136.
[42]Y. He and A. Wang. A simplified two-level method for the steady Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg.,197(2008),1568-1576.
[43]Y. He, Y. Zhang, Y. Shang, and H. Xu. Two-Level Newton Iterative Method for the 2D/3D Steady Navier-Stokes Equations. Numer. Methods Partial Differential Eq.,28(2012),1620-1642.
[44]F. Hua. Analysis and Simulation of Stokes-Darcy System with Beavers-Joseph Interface Condition:PhD, Florida State University, USA,2009.
[45]P. S. Huyakorn and G. F. Pinder. Computational methods in subsurface flow. Academic Press, New York,1983.
[46]S. Lei. An analytical solution for steady state flow into a tunnel. Ground Water,37(1999),23-26.
[47]P. Lesaint and M. Zlamal. Convergence of the Nonconforming Wilson Element for Arbitrary Quadrilateral Meshes. Numer. Math.,36(1980),33-52.
[48]R. Li, Z. Chen, and W. Wu. Generalized Difference Methods for Differential Equations Numerical Analysis of Finite Volume Methods. Marcel Dekker Inc. New York,2000.
[49]R. Liedl, M. Sauter, D. Huckinghaus, T. Clemens, and G. Teutsch. Simulation of the development of karst aquifers using a coupled continuum pipe flow model. Water Resour. Res.,39(2003), SBH6.1-SBH6.11.
[50]W. Liu, Z. Kang, H. Rui. Finite volume element approximation of the cou-pled continuum pipe-flow/Darcy model for flows in Karst aquifers. Numer. Methods Partial Differential Eq.,30(2014),376-392.
[51]W. Liu, H. Rui, and H. Guo. A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations. Acta Mathematicae Appli-catae Sinica,27(2011),495-502.
[52]W. Liu, H, Rui, and F. Hu. A two-grid algorithm for expanded mixed fi-nite element approximations of semi-linear elliptic equations. Computers and Mathematics with Applications,66(2013),392-402.
[53]H. Lopez, B. Molina, and J.S.Jose. Comparison between different numeri-cal discretizations for a Darcy-Forchheimer model. Electron. Trans. Numer. Anal.,34(2009),187-203.
[54]P. Luo and Q. Lin. High accuracy analysis of the Wilson element. J. Comp. Appl. Math.,17(1999),113-124.
[55]K. T. B. MacQuarrie and E. A. Sudicky. On the incorporation of drains into three-dimensional variably saturated groundwater flow models. Water Resour. Res.,32(1996),447-482.
[56]M. Marion and J. Xu. Error estimates on a new nonlinear Galerkin method based on two-grid fifite elements. SIAM J. Numer. Anal.,32(1995),1170-1184.
[57]F. A. Milner. Mixed finite element methods for quasilinear second order elliptic problems. Math. Comp.,44(1985),303-320.
[58]J. Murray. Mathematical biology, second ed.. Springer-Verlag, New York, 1993.
[59]J. C. Nedelec. Mixed finite element in R3. Numer. Math.,35(1980),315-341.
[60]S. P. Neuman. Theoretical derivation of Darcy's law. Acta Mechanica, 25(1977),153-170.
[61]R. A. Nicolaides, T. A. Porsching, and C. A. Hall. Covolume methods in computational fluid dynamics. Computational Fluid Dynamics Review, John Wiley and Sons, New York, (1995) 279-299.
[62]H. Pan and H. Rui. Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput.,52(2012),563-587.
[63]E. J. Park. Mixed finite element method for generalized Forchheimer flow in porous media. Numer. Methods Partial Differential Eq.,21(2005),213-228.
[64]V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere, McGraw-Hill, New York,1980.
[65]P. A. Raviart and J. M. Thomas. A mixed finite element method for 2nd order elliptic problems, in Proc. Conf. on Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Springer-Verlag, Berlin, vol.606, 1977,292-315.
[66]H. Rui and H. Pan. A block-centered finite difference method for the Darcy-Forchheimer model. SIAM J. Numer. Anal.,5(2012),2612-2631.
[67]D. Ruth and H. Ma. On the derivation of the Forchheimer equation by means of the averaging theorem. Transport in Porous Media,7(1992),255-264.
[68]M. Sauter. Quantication and forecasting of regional groundwater flow and transport in a karst aquifer (Gal lusquel le, Malm SW Germany), Ph.D. Thesis, Univ. of Tuingen, Tubinger, Germany,1992.
[69]Z.C. Shi and S. Chen. Analysis of a nine degree plate bending element of Shecht. Chinese J. Num. Math. Appl,4(1989),73-79.
[70]Z.C. Shi, B. Jiang and W. Xue. A new superconvergence property of Wilson nonconforming finite element. Numer. Math.,78(1997),259-268.
[71]D. Shi and H. Liang. Superconvergence analysis of Wilson element on anisotropic meshes. Applied Mathematics and Mechanics (English Edition), 28(2007),119-125.
[72]G. Teutsch. Two practical examples from the Swabian Alb, S. Germany, in Proc.4th Conference on Solving Groundwater Problems with Models, Indi-anapolis,1989.
[73]G. Teutsch and M. Sauter. Distributed parameter modeling approaches in karst-hydrological investigations. Bull Hydrogeol,16(1998),99-109.
[74]R. Therrien and E. A. Sudicky. Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media. J Contam. Hydrol.,23(1995),1-44.
[75]X. Wang. On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer. Discrete Contin. Dyn. Syst. Ser. B,13(2010),489-501.
[76]王云,于青春,薛亮,马浩.裂隙岩溶含水系统溢流泉演化过程的数值模拟.中国岩溶,第29卷,第4期,378-384页,2010.
[77]A. Weiser and M. F. Wheeler. On convergence of block-centered finite differ-ence for elliptic problems. SIAM J. Numer. Anal,25(1988),351-375.
[78]E.L. Wilson, R.L. Taylor, W.P. Doherty, and J. Ghaboussi. Incompatible dis-placement methods, in:S.J. Fenves (Ed.), Numerical and Computer Methods in Structural Mechanics, Academic Press, New York,1973.
[79]S. Whitaker. Flow in porous media I. A theoretical derivation of Darcy's law. Transport in Porous Media,1(1986),3-25.
[80]C. S. Woodward and C. N. Dawson. Analysis of expanded mixed finite ele-ment methods for a nonlinear parabolic equation modelling flow into variably saturated porous media. SIAM J. Numer. Anal,37(2000),701-724.
[81]L. Wu and M.B. Allen. A two grid method for mixed element finite solu-tion of reaction-diffusion equations. Numer. Methods Partial Differential Eq., 15(1999),317-332.
[82]H. Wu and R. Li. Error Estimates for Finite Volume Element Methods for General Second-Order Elliptic Problems. Numer. Methods Partial Differential Eq.,19(2003),693-708.
[83]J. Xu. A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput.,15(1994),231-237.
[84]J. Xu. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal.,33(1996),1759-1777.
[85]袁道先.新形势下我国岩溶研究面临的机遇和挑战.中国岩溶,第28卷,第4期,329-331页,2009.
[86]A. Zenisek and M. Vanmaele. The interpolation theorem for narrow quadri-lateral isoparametric finite elements. Numer. Math.,72(1995),123-141.
[2]T. Arbogast, M. F. Wheeler, and I. Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal,34(1997),828-852.
[3]K. Aziz and A. Settari. Petroleum Reservoir Simulation. Applied Science Publishers LTD, London,1979.
[4]R. E. Bank and D. J. Rose. Some error estimates for the box method. SIAM J. Numer. Anal.,24(1987),777-787.
[5]G. Barenblatt, I. Zheltov, and I. Kochina. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech., 24(1960),1286-1303.
[6]S. Bauer, R. Liedl, and M. Sauter. Modeling of karst aquifer genesis:influence of exchange flow. Water Resour. Res.,39(2003), SBH6.1-SBH6.12.
[7]S. Bauer, R. Liedl, and M. Sauter. Modelling of karst development consider-ing conduit-matrix exchange flow, Calibration and reliability in groundwater modelling:coping with uncertainty. IAHS Publ,265(2000),10-15.
[8]J. Bear and A. Verruijt. Modeling groundwater flow and pollution. D. Reidel Pub Co, Norwell, MA,1987.
[9]C. Bi and V. Ginting. Two-grid finite volume element for linear and nonlinear elliiptic problems. Numer. Math.,108(2007),177-198.
[10]S. Birk, R. Liedl, M. Sauter, and G. Teutsch. Hydraulic boundary conditions as a controlling factor in karst genesis. Water Resour. Res.,39(2003), SBH2.1-SBH2.14.
[11]F. Brezzi, J. Douglas Jr., M. Fortin, and L. Marini. Two families of mixed finite elements for second order elliptic problems. Numer. Math.,47(1985), 217-235.
[12]E. Bobok. Fluid mechanics for petroleum engineers. Elsevier Sci, New York, 1993.
[13]Z. Cai. On the finite volume element method. Numer. Math.,28(1991), 392-402.
[14]Y. Cao, M. Gunzburger, F. Hua, and X. Wang. Analysis and Finite Element Approximation of a Coupled, Continuum Pipe-Flow/Darcy Model for Flow in Porous Media with Embedded Conduits. Numer. Methods Partial Differential Eq.,27(2011),1242-1252.
[15]Z. Chen. The control volume finite element methods and their applications to multiphase flow. Networks and Heterogeneous Media,1(2006),689-706.
[16]Z. Chen. Expanded mixed element methods for linear second-order elliptic problems(Ⅰ). RAIRO Model. Math. Anal. Numer.,32(1998),479-499.
[17]Z. Chen. Expanded mixed element methods for quasilinear second-order el-liptic problems(Ⅱ). RAIRO Model. Math. Anal. Numer.,32(1998),501-520.
[18]Z. Chen. BDM mixed methods for a nonlinear elliptic problem. J. Comp. Appl. Math.,53(1994),207-223.
[19]L. Chen and Y. Chen. Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods. J Sci. Comput.,49(2011), 383-401.
[20]N. Chen, M. Gunzburger, B. Hu, X. Wang, and C. Woodruff. Calibrating the exchange coefficient in the modified coupled continuum pipe-flow model for flows in karst aquifers. J. Hydrol,414-415(2012),294-301.
[21]Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, in the Computational Science and Engineering Series, Vol. 2, SIAM, Philadelphia,2006.
[22]Y. Chen, Y. Huang, and D. Yu. A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int J. Numer. Meth. Eng.,57(2003),193-209.
[23]H. Chen and B. Li. Superconvergence analysis and error expansion for the Wilson nonconforming finite element. Numer. Math.,69(1994),125-140.
[24]C. Chen and W. Liu, Two-grid finite volume element methods for semilinear parabolic problems. Appl. Numer. Math.,60(2010),10-18.
[25]C. Chen and W. Liu. A two-grid method for finite volume element approxima-tions of second-order nonlinear hyperbolic equations, J. Comp. Appl. Math., 233(2010),2975-2984.
[26]Y. Chen, H. Liu, and S. Liu. Analysis of two-grid methods for reaction diffusion equations by expanded mixed finite element methods. Int J. Numer. Meth. Eng.,69(2007),408-422.
[27]S. Chen, D. Shi, and Y. Zhao. Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes. IMA Journal of Numerical Analysis, 24(2004),77-95.
[28]S. Chen, Y. Zhao, and D. Shi. Anisotropic interpolations with application to nonconforming elements. Applied Numerical Mathematics,49(2004),135-152.
[29]S. H. Chou, D. Y. Kwak, and Q. Li. LP error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differential Eq.,19(2003),463-486.
[30]S. H. Chou and Q. Li. Error estimates in L2, H1 and L∞ in covolume methods for elliptic and parabolic problems:A unified aproach. Math. Comp., 69(2000),103-120.
[31]T. Clemens, D. Huckinghaus, M. Sauter, R. Liedl, and G. Teutsch. A com-bined continuum and discrete network reactive transport model for the sim-ulation of karst development. IAHS Publ,237(1996),309-318.
[32]C. N. Dawson and M. F. Wheeler. Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. Contempt Maths,180(1994), 191-203.
[33]C. N. Dawson, M. F. Wheeler and C. S. Woodward. A two-grid finite dif-ference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal., 35(1998),435-452.
[34]R. E. Ewing, Z. Li, T. Lin, and Y. P. Lin. The immersed finite volume ele-ment methods for the elliptic interface problems. Math. Comput. Simulations, 50(1999),63-76.
[35]R. E. Ewing, T. Lin, and Y. P. Lin. On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal.,39(2002),1865-1888.
[36]P. FORCHHEIMER. Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing., 45(1901),1782-1788.
[37]V. Girault and M. F. Wheeler. Numerical discretization of a Darcy-Forchheimer model. Numer. Math.,110(2008),161-198.
[38]A. Harbaugh. The U.S. Geological survey modular groundwater model-the GroundWater Flow Process. U.S. Geological Survey Techniques and Methods 6-A16 (2005).
[39]Y. He and J. Li. Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg.,198(2009),1351-1359.
[40]Y. He and W. Sun. Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations. SIAM J.Numer. Anal.,45(2007),837-869.
[41]Y. He and W. Sun. Stabilized finite element methods based on Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comp.,76(2007),115-136.
[42]Y. He and A. Wang. A simplified two-level method for the steady Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg.,197(2008),1568-1576.
[43]Y. He, Y. Zhang, Y. Shang, and H. Xu. Two-Level Newton Iterative Method for the 2D/3D Steady Navier-Stokes Equations. Numer. Methods Partial Differential Eq.,28(2012),1620-1642.
[44]F. Hua. Analysis and Simulation of Stokes-Darcy System with Beavers-Joseph Interface Condition:PhD, Florida State University, USA,2009.
[45]P. S. Huyakorn and G. F. Pinder. Computational methods in subsurface flow. Academic Press, New York,1983.
[46]S. Lei. An analytical solution for steady state flow into a tunnel. Ground Water,37(1999),23-26.
[47]P. Lesaint and M. Zlamal. Convergence of the Nonconforming Wilson Element for Arbitrary Quadrilateral Meshes. Numer. Math.,36(1980),33-52.
[48]R. Li, Z. Chen, and W. Wu. Generalized Difference Methods for Differential Equations Numerical Analysis of Finite Volume Methods. Marcel Dekker Inc. New York,2000.
[49]R. Liedl, M. Sauter, D. Huckinghaus, T. Clemens, and G. Teutsch. Simulation of the development of karst aquifers using a coupled continuum pipe flow model. Water Resour. Res.,39(2003), SBH6.1-SBH6.11.
[50]W. Liu, Z. Kang, H. Rui. Finite volume element approximation of the cou-pled continuum pipe-flow/Darcy model for flows in Karst aquifers. Numer. Methods Partial Differential Eq.,30(2014),376-392.
[51]W. Liu, H. Rui, and H. Guo. A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations. Acta Mathematicae Appli-catae Sinica,27(2011),495-502.
[52]W. Liu, H, Rui, and F. Hu. A two-grid algorithm for expanded mixed fi-nite element approximations of semi-linear elliptic equations. Computers and Mathematics with Applications,66(2013),392-402.
[53]H. Lopez, B. Molina, and J.S.Jose. Comparison between different numeri-cal discretizations for a Darcy-Forchheimer model. Electron. Trans. Numer. Anal.,34(2009),187-203.
[54]P. Luo and Q. Lin. High accuracy analysis of the Wilson element. J. Comp. Appl. Math.,17(1999),113-124.
[55]K. T. B. MacQuarrie and E. A. Sudicky. On the incorporation of drains into three-dimensional variably saturated groundwater flow models. Water Resour. Res.,32(1996),447-482.
[56]M. Marion and J. Xu. Error estimates on a new nonlinear Galerkin method based on two-grid fifite elements. SIAM J. Numer. Anal.,32(1995),1170-1184.
[57]F. A. Milner. Mixed finite element methods for quasilinear second order elliptic problems. Math. Comp.,44(1985),303-320.
[58]J. Murray. Mathematical biology, second ed.. Springer-Verlag, New York, 1993.
[59]J. C. Nedelec. Mixed finite element in R3. Numer. Math.,35(1980),315-341.
[60]S. P. Neuman. Theoretical derivation of Darcy's law. Acta Mechanica, 25(1977),153-170.
[61]R. A. Nicolaides, T. A. Porsching, and C. A. Hall. Covolume methods in computational fluid dynamics. Computational Fluid Dynamics Review, John Wiley and Sons, New York, (1995) 279-299.
[62]H. Pan and H. Rui. Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput.,52(2012),563-587.
[63]E. J. Park. Mixed finite element method for generalized Forchheimer flow in porous media. Numer. Methods Partial Differential Eq.,21(2005),213-228.
[64]V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere, McGraw-Hill, New York,1980.
[65]P. A. Raviart and J. M. Thomas. A mixed finite element method for 2nd order elliptic problems, in Proc. Conf. on Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Springer-Verlag, Berlin, vol.606, 1977,292-315.
[66]H. Rui and H. Pan. A block-centered finite difference method for the Darcy-Forchheimer model. SIAM J. Numer. Anal.,5(2012),2612-2631.
[67]D. Ruth and H. Ma. On the derivation of the Forchheimer equation by means of the averaging theorem. Transport in Porous Media,7(1992),255-264.
[68]M. Sauter. Quantication and forecasting of regional groundwater flow and transport in a karst aquifer (Gal lusquel le, Malm SW Germany), Ph.D. Thesis, Univ. of Tuingen, Tubinger, Germany,1992.
[69]Z.C. Shi and S. Chen. Analysis of a nine degree plate bending element of Shecht. Chinese J. Num. Math. Appl,4(1989),73-79.
[70]Z.C. Shi, B. Jiang and W. Xue. A new superconvergence property of Wilson nonconforming finite element. Numer. Math.,78(1997),259-268.
[71]D. Shi and H. Liang. Superconvergence analysis of Wilson element on anisotropic meshes. Applied Mathematics and Mechanics (English Edition), 28(2007),119-125.
[72]G. Teutsch. Two practical examples from the Swabian Alb, S. Germany, in Proc.4th Conference on Solving Groundwater Problems with Models, Indi-anapolis,1989.
[73]G. Teutsch and M. Sauter. Distributed parameter modeling approaches in karst-hydrological investigations. Bull Hydrogeol,16(1998),99-109.
[74]R. Therrien and E. A. Sudicky. Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media. J Contam. Hydrol.,23(1995),1-44.
[75]X. Wang. On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer. Discrete Contin. Dyn. Syst. Ser. B,13(2010),489-501.
[76]王云,于青春,薛亮,马浩.裂隙岩溶含水系统溢流泉演化过程的数值模拟.中国岩溶,第29卷,第4期,378-384页,2010.
[77]A. Weiser and M. F. Wheeler. On convergence of block-centered finite differ-ence for elliptic problems. SIAM J. Numer. Anal,25(1988),351-375.
[78]E.L. Wilson, R.L. Taylor, W.P. Doherty, and J. Ghaboussi. Incompatible dis-placement methods, in:S.J. Fenves (Ed.), Numerical and Computer Methods in Structural Mechanics, Academic Press, New York,1973.
[79]S. Whitaker. Flow in porous media I. A theoretical derivation of Darcy's law. Transport in Porous Media,1(1986),3-25.
[80]C. S. Woodward and C. N. Dawson. Analysis of expanded mixed finite ele-ment methods for a nonlinear parabolic equation modelling flow into variably saturated porous media. SIAM J. Numer. Anal,37(2000),701-724.
[81]L. Wu and M.B. Allen. A two grid method for mixed element finite solu-tion of reaction-diffusion equations. Numer. Methods Partial Differential Eq., 15(1999),317-332.
[82]H. Wu and R. Li. Error Estimates for Finite Volume Element Methods for General Second-Order Elliptic Problems. Numer. Methods Partial Differential Eq.,19(2003),693-708.
[83]J. Xu. A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput.,15(1994),231-237.
[84]J. Xu. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal.,33(1996),1759-1777.
[85]袁道先.新形势下我国岩溶研究面临的机遇和挑战.中国岩溶,第28卷,第4期,329-331页,2009.
[86]A. Zenisek and M. Vanmaele. The interpolation theorem for narrow quadri-lateral isoparametric finite elements. Numer. Math.,72(1995),123-141.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |