多孔介质渗流问题守恒特征线数值方法及理论
|
摘要
多孔介质中的流体运动方程广泛应用于地下水、环境科学和油藏模拟等领域[11,62].模型主要包括了流体的流动和质量的转移,体现着流体本身的质量守恒、能量守恒等物理性质.其中重力、粘度、毛细管力、密度等对该物理过程都起着重要作用,而工程上考虑的重要参数一般包括流体速度、压力、温度以及浓度等.数学上该模型由描述多孔介质中的速度方程即Darcy定律和描述组分混溶传质过程的浓度方程构成.基于一些合理的假设条件可以将方程简化,但仍表现为依赖于时间的非线性耦合问题.对此模型研究保持物理性质的高效数值模拟格式有着重要的实际意义和价值.
渗流的速度方程通常由Darcy定律描述[26,45],它体现了流体的平均速度与压力梯度的线性关系.由于压力和速度的重要性,对不可压缩条件下的质量守恒方程和Darcy方程构成的模型一般采用的是混合元离散格式,这样即保证速度和压力的精度,又能保证局部的质量守恒.经典的混合元空间Raviart-Thomas、Brezzi-Douglas-Marini等保证了速度法向分量的连续性,并且都给出了解的存在唯一性及最优的误差估计[17,18,19,65].大量的稳定化技术也可以用来求解Darcy问题.例如Masud和Hughes[60]加入的稳定项使得连续的速度压力都是有效的,当然还有加一个平方项或者引入Bubble函数等稳定化技术.有时需要速度是连续的,也就是切向方向速度也是连续的,Arbogast和Wheeler[6]给出了一类连续速度通量的逼近格式,虽然损失了散度范数的最优收敛率,但还是得到了最优的L2模误差估计.
Darcy流与Stokes流耦合问题在实际中也有着广泛应用.即在多孔介质的基础上耦合一个自由流区域的Stokes问题,两个子问题内边界由Beavers-Joseph-Saffman条件确定[72],这两个子问题的正则性要求不一样,并且内边界上切线速度是不连续的Layton等人[56]详细介绍了该模型,并引入Lagrange乘子证明了弱解的存在唯一性Yotov[66]给出了一类求解技术,在两个子区域采用不同的离散空间,即在Stokes区域上用DG求解,而Darcy区域用混合元技术求解.这样不利于处理内边界条件和编程.而相同网格剖分下的同一种混合元离散无疑是高效的,Burman[21]给出非协调的Crouzeix-Raviart有限元空间逼近格式.芮和张[69]对其改进,引入了更简单的罚项,保证稳定的基础上还有局部质量守恒性Arbogast和Brunson[3]还将[6]提出的连续速度空间用到该耦合问题中,并给出了最优的L2误差估计.对于奇异扰动问题即Brinkman方程,Mardal[59]给出了该问题弱形式解的存在唯一性.提出了一种绝对稳定的有限元格式,并对Brinkman问题给出了系统的理论分析和最优的误差估计.在此基础上后面陆续有很多的离散格式[61,83].
而组分的混溶传质过程最终可以由一个关于浓度的对流扩散方程所表示.当对流占优时,由于数值振荡和物理弥散的效应,传统的有限元和有限差分在进行模拟时效果并不理想,此时特征线技术很好的解决了这一难题.该技术从数学上将对流扩散问题转化一个等价的易处理的扩散问题,体现出很好的优越性,并且对于时间步长不再有很强的限制,用来模拟大时间步长的实际问题时效果明显.该思想早期是由[43]提出的向前追踪特征线方法,但是这样的技术破坏了原始空间剖分,存在很大的局限性Douglas和Russell [54]1982年提出了向后追踪的修正特征线技术(MMOC),给出基于有限元和有限差分方法下的离散格式,得到有限元离散下最优的H1与L2模估计.此后该技术迅速得到广泛应用,Russell [71]将其运用到了不可压缩的混溶驱动模型中去,压力方程采取有限元离散,并给出了最优阶的误差估计Ewing等[35,36]也采用MMOC技术求解浓度方程,但对压力方程采用更加有效的混合元技术,使得速度、压力和浓度均能达到最优估计.此外在模拟对流扩散问题时,一个重要的性质就是要保证质量守恒,也就是组分的总量在不考虑源汇项的情况下随时间是平稳不变的,上述技术并不能满足这一点.芮[68]提出了对流扩散问题的质量守恒格式,此时的速度有连续性的要求Celia[22]给出了ELLAM技术保证了质量守恒,但是计算有些复杂,Wang[77,78]给出了ELLAM格式在对流扩散问题中的收敛性及最优估计,并给出了一些实际模拟.对于更为复杂的可压缩混溶驱动的模型Douglas和Roberts [53]给出可压缩混溶驱动的数学模型,并给出了基于有限元和混合元方法的半离散格式的误差估计,此后陈[23]给出了混溶驱动的全离散数值格式.袁和程也分别给出了基于MMOC的有限元误差估计[24,85].
韩和吴[47]对于Stokes方程提出了基于交错网格的混合元离散技术,即速度分量和压力使用三套不同的网格剖分,数值模拟更加方便.我们在此基础上加以修改得到对Darcy方程适用的连续速度逼近.参照对流扩散问题的守恒格式[68],得到渗流中的不可压缩混溶驱动模型的质量守恒的特征线数值格式及误差分析,随后给出了可压缩混溶驱动模型的全离散数值格式及误差分析,并分别给出数值实验进行验证.
本文的组织结构如下：
在第一章中,介绍了多孔介质渗流问题的数学模型.基于流体的物理性质给出Darcy定律及质量守恒方程.结合状态方程给出了混溶驱动中组分传质的质量守恒方程,得到所谓的浓度方程.同样也给出了可压缩流体的浓度方程.给出了常用的函数空间的记号和范数定义,最后给出了本文理论推导需要的一些不等式.
在第二章中,对于Darcy司题,给出了基于交错网格剖分下的连续速度逼近.借助RT空间,得到很好的插值性质.给出了数值格式的误差分析.虽然得不到散度范数的误差,但还是给出了L2误差估计.数值算例中与Arbogast和Wheeler[6]中提到的混合元空间进行了比较,数值结果接近,但是用的自由度相对较少,这样可以减少时间复杂度.
在第三章中,首先用连续元求解稳态的Dracy-Stokes耦合问题,给出了全离散格式及误差分析,数值实验验证了收敛性.第二部分考虑了扰动的Darcy-Stokes问题,即Brinkman模型,误差分析得到L2误差估计,数值实验验证了未知量受扰动因子影响下数值逼近的收敛率.
在第四章中,给出了多孔介质中不可压缩混溶驱动的守恒特征线数值格式.运用[68]提出的守恒特征线格式,得到混溶驱动的守恒特征线技术(MCC),理论证明了数值格式的质量守恒性.利用外插技术,将速度方程和浓度方程解耦,在一些归纳假设的条件下给出离散格式的误差分析和最优的L2模误差估计.数值算例验证了该逼近格式的质量守恒性,未知量的收敛阶也是与理论一致的.最后给出了一个实际问题的模拟,验证了数值格式的有效性.
在第五章中,讨论了多孔介质可压缩混溶驱动模型的特征线数值格式.该模型为较强的非线性耦合问题,利用投影算子首先给出未知量的初始值,然后基于一些归纳假设进行误差分析,对速度方程和浓度方程对应的误差方程分别进行估计,再整合到一起最终得到最优的误差估计.最后用数值实验验证了理论分析.
The mathematical model for fluid flow in porous media is widely used in many scientific fields including groundwater,environment science, reservoir simulation and so on[11,62]. This model includes the fluid motion and mass transfer, which are based on conservation of mass, momentum and energy. Gravity, viscous, capillary and density play important role in this process, but we take more attention to fluid velocity, pressure, temperature and con-centration in reservoir simulation and engineering. The flow motion of fluid is described by Darcy's law and the mass transfer is described by concen-tration equation. We can simplify these equations under some reasonable assumption, but it is still shown to be a nonlinear complex coupled system. It is valuable and significant to propose effective numerical scheme to keep physical properties of this system.
The analytical model describing fluid flow in porous media is Darcy's law[26,45], which shows the linear relation between average fluid velocity and pressure gradient. Mixed finite element scheme can be used to solve this model which coupled the Darcy's law and mass conservation equation, as the velocity and pressure can be approximated accurately, simultaneously local mass conservation is kept. The classical Raviart-Thomas, Brezzi-Douglas-Marini mixed finite element methods keep the continuity of the approxi-mate velocity in the normal direction, and the existence, uniqueness and the optimal error estimate of discrete solution are obtained[17,18,19,65]. Some stabilization techniques can be used to solve this model, Masud and Hughes [60] added stabile term to the equation so that all continuous con-forming velocity-pressure spaces are viable. Least-square formulation and bubble function can also be used for stabilization. In some application, it is desirable that the velocity in the tangential direction is continuous. Arbo-gast and Wheeler [6] give a continuous velocity flux, though it loss the optimal convergence of the divergence norm, the L2error is optimal.
We also consider coupling model which means that Darcy equation cou-pled anther Stokes equation in fluid region. They coupled with an Beavers-Joseph-Saffman condition[72] across the interface. The two parts have dif-ferent regularity properties, and the tangential velocity is discontinuous on the interface. Latyon[56] introduces the mathematical model, give the proof of existence of weak solutions by using Lagrange multipliers.[66] Yotov solve this coupled model with different finite methods on the two regions, they solve Stokes equation with DG method and Darcy equation with mixed fi-nite element. But it is not conducive difficulty to deal with the interface and programming. Burman[21] give the nonconforming stabilized Crouzeix-Raviart element, then Zhang[69] improve this scheme, and give a simpler stabilization penalty term to maintain element-wise mass conservation. Ar-bogast and Brunson[3] solve this coupled system refer to[6], give optimal L2convergence of velocity and pressure. Brinkman equation can be viewed as singular perturbation problem, Mardal[59] give the proof of existence and uniqueness of its weak solutions, introduce a robust finite element spaces, and gives the error estimate in detail. Following this analysis, more useful discrete schemes are discussed [61,83].
Mathematically, the process of miscible displacement is described by a convection diffusion equation for the concentration of each chemical compo-nent. In the case of convection-dominated, classical finite element and finite difference methods do not work well because of the process of diffusion and dispersion. And the method of characteristics is efficient to treat this diffi-culty, as it transform primal equation to equivalent diffusion equation and have no strong constraint to time step. So we can use this skill to solve practical problem which need big time step. It is constructed initially as for-ward tracking method of characteristic [43], but it destroys the original space mesh with strong limitation. Douglas and Russell[54] introduce back tracking method of characteristic (modified method of characteristics MMOC) which overcome the restriction above. They combine finite element and finite differ-ence methods, moreover the optimal error estimates in L2, H1are derived for finite element scheme then this skill is widely used rapidly. Russell[71] apply it to incompressible miscible displacement, which Galerkin method is used to approximate the pressure, and optimal L2,H1convergence rates are given. In [35,36] MMOC is used to solve concentration equation and mixed finite ele-ment to pressure equation, in this way all the unknowns can be approximate optimally. In addition, for convection diffusion equation, an important prop-erty is mass conservation, which means that the total mass of component is balance along time direction regardless of sink and source, but general char-acteristics can't meet it. Rui[68] construct a new characteristics scheme for convection diffusion equation which maintains mass conservation, which need continuity velocity. Celia[22] propose euler-lagrange localized adjoint method (ELLAM) scheme which is mass-balance, but bring trouble to computation. Wang[77,78] give the optimal error estimate of ELLAM scheme for convec-tion diffusion equation, while show computational experiments to simulate practical problem. For more complex compressible miscible displacement, Douglas and Roberts [53]give the nonlinear parabolic system, approximate the pressure with both Galerkin and mixed finite element method and give corresponding error for semi-discrete scheme. Chen and Ewing [23] analyze its full discrete scheme with finite element method, Cheng and Yuan also give the error estimate of finite element scheme base on MMOC [24,85].
Han and Wu[47] give a mixed finite element with continue velocity flux on staggered mesh for Stokes equation, where the two components of the velocity and the pressure are defined on three different sets of grid-nodes. We modified it to obtain proper approximation for Darcy equation. Refer to [68], we obtain a new mass-balance scheme and error estimate for incompressible miscible displacement in porous media, and analyze the numerical scheme and error scheme for nonlinear coupled system that describes compressible miscible displacement. In this two cases, we all give numerical tests to verify our proof.
The outline of the thesis is as follows.
In Chapter1, we introduce the model which describes fluid flow in porous media, give Darcy's law and mass conservation equation base on physical properties. Combine with state equation we give concentration equation describing mass transfer process of compressible and incompressible miscible displacement. Then we introduce some definitions of function space and corresponding norm, and show serval useful inequalities for this paper.
In Chapter2, we introduce a mixed finite element method with contin-uous flux for an elliptic equation modeling Darcy flow in porous media. we present a better property of interpolation with the help of RT mixed finite element spaces and give convergence rate of L2norm. At last we give nu-merical examples to compare the new element with the mixed finite element space in [6]. We can see the convergence rates of this two scheme are almost identical, however we need less freedom of degree which need less time for computation.
In Chapter3, we solve coupled Darcy-Stokes model with continue veloc-ity flux, give the error estimates and verify the convergence rate by numeri-cal experiments, then we consider Brinkman equation which is perturbation problem, we get convergence rate of L2norm. The numerical example val-idate the convergence rate of the unknowns which depend on perturbation parameter.
In Chapter4, we analyze a numerical scheme for the coupled system of incompressible miscible displacement in porous media. Mass-conservative characteristic finite element is used for concentration equation refer to [68]. We prove the property of mass balance and decouple the velocity equation and concentration equation base on extrapolation. Under some reasonable assumption we obtain error estimate and convergence rate of L2norm. We also verify the mass balance and convergence rate by numerical examples. At last we give experiment of practical problem.
In Chapter5, we discuss a numerical scheme for the coupled system of compressible miscible displacement in porous media, which is a strong non-linear and coupled system. Firstly we get the original value of the unknowns by projection. Secondly under some reasonable assumption we obtain error estimates for velocity and concentration equation respectively, then get the optimal convergence rate. Finally numerical examples verify the theoretical analysis.
渗流的速度方程通常由Darcy定律描述[26,45],它体现了流体的平均速度与压力梯度的线性关系.由于压力和速度的重要性,对不可压缩条件下的质量守恒方程和Darcy方程构成的模型一般采用的是混合元离散格式,这样即保证速度和压力的精度,又能保证局部的质量守恒.经典的混合元空间Raviart-Thomas、Brezzi-Douglas-Marini等保证了速度法向分量的连续性,并且都给出了解的存在唯一性及最优的误差估计[17,18,19,65].大量的稳定化技术也可以用来求解Darcy问题.例如Masud和Hughes[60]加入的稳定项使得连续的速度压力都是有效的,当然还有加一个平方项或者引入Bubble函数等稳定化技术.有时需要速度是连续的,也就是切向方向速度也是连续的,Arbogast和Wheeler[6]给出了一类连续速度通量的逼近格式,虽然损失了散度范数的最优收敛率,但还是得到了最优的L2模误差估计.
Darcy流与Stokes流耦合问题在实际中也有着广泛应用.即在多孔介质的基础上耦合一个自由流区域的Stokes问题,两个子问题内边界由Beavers-Joseph-Saffman条件确定[72],这两个子问题的正则性要求不一样,并且内边界上切线速度是不连续的Layton等人[56]详细介绍了该模型,并引入Lagrange乘子证明了弱解的存在唯一性Yotov[66]给出了一类求解技术,在两个子区域采用不同的离散空间,即在Stokes区域上用DG求解,而Darcy区域用混合元技术求解.这样不利于处理内边界条件和编程.而相同网格剖分下的同一种混合元离散无疑是高效的,Burman[21]给出非协调的Crouzeix-Raviart有限元空间逼近格式.芮和张[69]对其改进,引入了更简单的罚项,保证稳定的基础上还有局部质量守恒性Arbogast和Brunson[3]还将[6]提出的连续速度空间用到该耦合问题中,并给出了最优的L2误差估计.对于奇异扰动问题即Brinkman方程,Mardal[59]给出了该问题弱形式解的存在唯一性.提出了一种绝对稳定的有限元格式,并对Brinkman问题给出了系统的理论分析和最优的误差估计.在此基础上后面陆续有很多的离散格式[61,83].
而组分的混溶传质过程最终可以由一个关于浓度的对流扩散方程所表示.当对流占优时,由于数值振荡和物理弥散的效应,传统的有限元和有限差分在进行模拟时效果并不理想,此时特征线技术很好的解决了这一难题.该技术从数学上将对流扩散问题转化一个等价的易处理的扩散问题,体现出很好的优越性,并且对于时间步长不再有很强的限制,用来模拟大时间步长的实际问题时效果明显.该思想早期是由[43]提出的向前追踪特征线方法,但是这样的技术破坏了原始空间剖分,存在很大的局限性Douglas和Russell [54]1982年提出了向后追踪的修正特征线技术(MMOC),给出基于有限元和有限差分方法下的离散格式,得到有限元离散下最优的H1与L2模估计.此后该技术迅速得到广泛应用,Russell [71]将其运用到了不可压缩的混溶驱动模型中去,压力方程采取有限元离散,并给出了最优阶的误差估计Ewing等[35,36]也采用MMOC技术求解浓度方程,但对压力方程采用更加有效的混合元技术,使得速度、压力和浓度均能达到最优估计.此外在模拟对流扩散问题时,一个重要的性质就是要保证质量守恒,也就是组分的总量在不考虑源汇项的情况下随时间是平稳不变的,上述技术并不能满足这一点.芮[68]提出了对流扩散问题的质量守恒格式,此时的速度有连续性的要求Celia[22]给出了ELLAM技术保证了质量守恒,但是计算有些复杂,Wang[77,78]给出了ELLAM格式在对流扩散问题中的收敛性及最优估计,并给出了一些实际模拟.对于更为复杂的可压缩混溶驱动的模型Douglas和Roberts [53]给出可压缩混溶驱动的数学模型,并给出了基于有限元和混合元方法的半离散格式的误差估计,此后陈[23]给出了混溶驱动的全离散数值格式.袁和程也分别给出了基于MMOC的有限元误差估计[24,85].
韩和吴[47]对于Stokes方程提出了基于交错网格的混合元离散技术,即速度分量和压力使用三套不同的网格剖分,数值模拟更加方便.我们在此基础上加以修改得到对Darcy方程适用的连续速度逼近.参照对流扩散问题的守恒格式[68],得到渗流中的不可压缩混溶驱动模型的质量守恒的特征线数值格式及误差分析,随后给出了可压缩混溶驱动模型的全离散数值格式及误差分析,并分别给出数值实验进行验证.
本文的组织结构如下：
在第一章中,介绍了多孔介质渗流问题的数学模型.基于流体的物理性质给出Darcy定律及质量守恒方程.结合状态方程给出了混溶驱动中组分传质的质量守恒方程,得到所谓的浓度方程.同样也给出了可压缩流体的浓度方程.给出了常用的函数空间的记号和范数定义,最后给出了本文理论推导需要的一些不等式.
在第二章中,对于Darcy司题,给出了基于交错网格剖分下的连续速度逼近.借助RT空间,得到很好的插值性质.给出了数值格式的误差分析.虽然得不到散度范数的误差,但还是给出了L2误差估计.数值算例中与Arbogast和Wheeler[6]中提到的混合元空间进行了比较,数值结果接近,但是用的自由度相对较少,这样可以减少时间复杂度.
在第三章中,首先用连续元求解稳态的Dracy-Stokes耦合问题,给出了全离散格式及误差分析,数值实验验证了收敛性.第二部分考虑了扰动的Darcy-Stokes问题,即Brinkman模型,误差分析得到L2误差估计,数值实验验证了未知量受扰动因子影响下数值逼近的收敛率.
在第四章中,给出了多孔介质中不可压缩混溶驱动的守恒特征线数值格式.运用[68]提出的守恒特征线格式,得到混溶驱动的守恒特征线技术(MCC),理论证明了数值格式的质量守恒性.利用外插技术,将速度方程和浓度方程解耦,在一些归纳假设的条件下给出离散格式的误差分析和最优的L2模误差估计.数值算例验证了该逼近格式的质量守恒性,未知量的收敛阶也是与理论一致的.最后给出了一个实际问题的模拟,验证了数值格式的有效性.
在第五章中,讨论了多孔介质可压缩混溶驱动模型的特征线数值格式.该模型为较强的非线性耦合问题,利用投影算子首先给出未知量的初始值,然后基于一些归纳假设进行误差分析,对速度方程和浓度方程对应的误差方程分别进行估计,再整合到一起最终得到最优的误差估计.最后用数值实验验证了理论分析.
The mathematical model for fluid flow in porous media is widely used in many scientific fields including groundwater,environment science, reservoir simulation and so on[11,62]. This model includes the fluid motion and mass transfer, which are based on conservation of mass, momentum and energy. Gravity, viscous, capillary and density play important role in this process, but we take more attention to fluid velocity, pressure, temperature and con-centration in reservoir simulation and engineering. The flow motion of fluid is described by Darcy's law and the mass transfer is described by concen-tration equation. We can simplify these equations under some reasonable assumption, but it is still shown to be a nonlinear complex coupled system. It is valuable and significant to propose effective numerical scheme to keep physical properties of this system.
The analytical model describing fluid flow in porous media is Darcy's law[26,45], which shows the linear relation between average fluid velocity and pressure gradient. Mixed finite element scheme can be used to solve this model which coupled the Darcy's law and mass conservation equation, as the velocity and pressure can be approximated accurately, simultaneously local mass conservation is kept. The classical Raviart-Thomas, Brezzi-Douglas-Marini mixed finite element methods keep the continuity of the approxi-mate velocity in the normal direction, and the existence, uniqueness and the optimal error estimate of discrete solution are obtained[17,18,19,65]. Some stabilization techniques can be used to solve this model, Masud and Hughes [60] added stabile term to the equation so that all continuous con-forming velocity-pressure spaces are viable. Least-square formulation and bubble function can also be used for stabilization. In some application, it is desirable that the velocity in the tangential direction is continuous. Arbo-gast and Wheeler [6] give a continuous velocity flux, though it loss the optimal convergence of the divergence norm, the L2error is optimal.
We also consider coupling model which means that Darcy equation cou-pled anther Stokes equation in fluid region. They coupled with an Beavers-Joseph-Saffman condition[72] across the interface. The two parts have dif-ferent regularity properties, and the tangential velocity is discontinuous on the interface. Latyon[56] introduces the mathematical model, give the proof of existence of weak solutions by using Lagrange multipliers.[66] Yotov solve this coupled model with different finite methods on the two regions, they solve Stokes equation with DG method and Darcy equation with mixed fi-nite element. But it is not conducive difficulty to deal with the interface and programming. Burman[21] give the nonconforming stabilized Crouzeix-Raviart element, then Zhang[69] improve this scheme, and give a simpler stabilization penalty term to maintain element-wise mass conservation. Ar-bogast and Brunson[3] solve this coupled system refer to[6], give optimal L2convergence of velocity and pressure. Brinkman equation can be viewed as singular perturbation problem, Mardal[59] give the proof of existence and uniqueness of its weak solutions, introduce a robust finite element spaces, and gives the error estimate in detail. Following this analysis, more useful discrete schemes are discussed [61,83].
Mathematically, the process of miscible displacement is described by a convection diffusion equation for the concentration of each chemical compo-nent. In the case of convection-dominated, classical finite element and finite difference methods do not work well because of the process of diffusion and dispersion. And the method of characteristics is efficient to treat this diffi-culty, as it transform primal equation to equivalent diffusion equation and have no strong constraint to time step. So we can use this skill to solve practical problem which need big time step. It is constructed initially as for-ward tracking method of characteristic [43], but it destroys the original space mesh with strong limitation. Douglas and Russell[54] introduce back tracking method of characteristic (modified method of characteristics MMOC) which overcome the restriction above. They combine finite element and finite differ-ence methods, moreover the optimal error estimates in L2, H1are derived for finite element scheme then this skill is widely used rapidly. Russell[71] apply it to incompressible miscible displacement, which Galerkin method is used to approximate the pressure, and optimal L2,H1convergence rates are given. In [35,36] MMOC is used to solve concentration equation and mixed finite ele-ment to pressure equation, in this way all the unknowns can be approximate optimally. In addition, for convection diffusion equation, an important prop-erty is mass conservation, which means that the total mass of component is balance along time direction regardless of sink and source, but general char-acteristics can't meet it. Rui[68] construct a new characteristics scheme for convection diffusion equation which maintains mass conservation, which need continuity velocity. Celia[22] propose euler-lagrange localized adjoint method (ELLAM) scheme which is mass-balance, but bring trouble to computation. Wang[77,78] give the optimal error estimate of ELLAM scheme for convec-tion diffusion equation, while show computational experiments to simulate practical problem. For more complex compressible miscible displacement, Douglas and Roberts [53]give the nonlinear parabolic system, approximate the pressure with both Galerkin and mixed finite element method and give corresponding error for semi-discrete scheme. Chen and Ewing [23] analyze its full discrete scheme with finite element method, Cheng and Yuan also give the error estimate of finite element scheme base on MMOC [24,85].
Han and Wu[47] give a mixed finite element with continue velocity flux on staggered mesh for Stokes equation, where the two components of the velocity and the pressure are defined on three different sets of grid-nodes. We modified it to obtain proper approximation for Darcy equation. Refer to [68], we obtain a new mass-balance scheme and error estimate for incompressible miscible displacement in porous media, and analyze the numerical scheme and error scheme for nonlinear coupled system that describes compressible miscible displacement. In this two cases, we all give numerical tests to verify our proof.
The outline of the thesis is as follows.
In Chapter1, we introduce the model which describes fluid flow in porous media, give Darcy's law and mass conservation equation base on physical properties. Combine with state equation we give concentration equation describing mass transfer process of compressible and incompressible miscible displacement. Then we introduce some definitions of function space and corresponding norm, and show serval useful inequalities for this paper.
In Chapter2, we introduce a mixed finite element method with contin-uous flux for an elliptic equation modeling Darcy flow in porous media. we present a better property of interpolation with the help of RT mixed finite element spaces and give convergence rate of L2norm. At last we give nu-merical examples to compare the new element with the mixed finite element space in [6]. We can see the convergence rates of this two scheme are almost identical, however we need less freedom of degree which need less time for computation.
In Chapter3, we solve coupled Darcy-Stokes model with continue veloc-ity flux, give the error estimates and verify the convergence rate by numeri-cal experiments, then we consider Brinkman equation which is perturbation problem, we get convergence rate of L2norm. The numerical example val-idate the convergence rate of the unknowns which depend on perturbation parameter.
In Chapter4, we analyze a numerical scheme for the coupled system of incompressible miscible displacement in porous media. Mass-conservative characteristic finite element is used for concentration equation refer to [68]. We prove the property of mass balance and decouple the velocity equation and concentration equation base on extrapolation. Under some reasonable assumption we obtain error estimate and convergence rate of L2norm. We also verify the mass balance and convergence rate by numerical examples. At last we give experiment of practical problem.
In Chapter5, we discuss a numerical scheme for the coupled system of compressible miscible displacement in porous media, which is a strong non-linear and coupled system. Firstly we get the original value of the unknowns by projection. Secondly under some reasonable assumption we obtain error estimates for velocity and concentration equation respectively, then get the optimal convergence rate. Finally numerical examples verify the theoretical analysis.
引文
[1]R. A. Adams and J. J. F. Fournier. Sobolev Spaces. Academic Press, Ams-terdam, second edition,2003.
[2]P. Angot. Analysis of singular perturbation on the brinkman problem for fictious domain models of viscous flows. Math. Methods Appl. Sci.,22:1395-1412,1999.
[3]T. Arbogast and D. D. Brunson. A computational method for approximating a darcy-stokes system governing a vuggy porous medium. Comput. Geosci., 11:207-218,2007.
[4]T. Arbogast and H. L. Lehr. Homogenization of a darcy-stokes system mod-eling vuggy porous media. Comput. Geosci.,10(3):291-302,2006.
[5]T. Arbogast and M. F. Wheeler. A characteristics-mixed finite element method for advection-dominated transport problems. SIAM J. Numer. Anal., 32:404-424,1995.
[6]T. Arbogast and M. F. Wheeler. A family of rectangular mixed element with a continuous flux for second order elliptic problems. SIAM J. Numer. Anal., 42:1914-1931,2005.
[7]I. Babuska. Error bounds for finite element methods. Numer. Math.,16:322-333,1971.
[8]L. Badea, M. Discacciati, and A.Quarteroni. Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Mi-lan,2006.
[9]S. Badia and R. Codina. Unified stabilized finite element formulations for the stokes and the darcy problems. SIAM J. Numer. Anal.,47(3):1971-2000, 2009.
[10]J. W. Barrett and K. W. Morton. Approximate symmetrization and petrov-galerkin methods for diffusion-convection problems. Comput. Methods Appl. Mech. Eng.,45:97-122,1984.
[11]J. Bear. Dynamics of Fluids in Porous Media. Elsevier, New York,1972.
[12]G. S. Beavers and D. D. Joseph. Boundary conditions at a naturally imper-meable wall. J. Fluid Mech.,30:197-207,1967.
[13]J. M. Boland and R. A. Nicolaides. On the stability of bilinear-constant velocity-pressure finite elements. Numer. Math.,44:219-222,1984.
[14]S. C. Brenner. Korn's inequalities for piecewise hl vector fields. Math. Com-put.,73:1067-1087,2003.
[15]S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York,1994.
[16]F. Brezzi. On the existence, uniqueness and approximation of saddle point problems arising from lagrangian multipliers. RAIRO Model. Math. Anal. Numer.,8:129-151,1974.
[17]F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, 1991.
[18]F. Brezzi. J. Douglas Jr., R. Duran, and M. Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer. Math.,51:237-250, 1987.
[19]F. Brezzi, J. Douglas Jr., M. Fortin, and L. D. Marini. Efficient rectangular mixed finite elements in two and three space variables. Math. Model. Numer. Anal.,21:581-604,1987.
[20]F. Brezzi, J. Douglas Jr., and L. D. Marini. Two families of mixed elements for second order ellptic problems. Numer. Math.,88:217-235,1985.
[21]E. Burman and P. Hansbo. Stabilized crouzeix-raviart element for the darcy-stokes problem. Numer Methods Partial Differ. Eq.,21:986-997,2005.
[22]M. A. Celia, T. F. Russell, I. Herrera, and R. E. Ewing. An eulerian-lagrangian localized adjoint method for the advection-diffusion equation. Ad-vances in Water Resources,13:187-206,1990.
[23]Z. Chen and R. E. Ewing. Fully discrete finite element analysis of multiphase flow in groundwater hydrology. SIAM J. Numer. Anal.,34:2228-253,1977.
[24]A. Cheng and G. Wang. The modified method of characteristics for com-pressible flow in porous media. Lecture Notes in Physics,552:69-79,2000.
[25]S. H. Chou and Q. Li. Mixed finite element methods for compressible miscible displacement in porous media. Mathematics of Computation,57(196):507-527,1991.
[26]P. G. Ciarlet. The Finite Element Method for Elliptic Problems. Amsterdam, North-Holland,1978.
[27]P. G. Ciarlet and P. A. Raviart. General lagrange and hermite interpolation in rn with applications to finite element methods. Arch. Rat. Mech. Anal., 46:177-199,1972.
[28]R. Codina. A stabilized finite element method for generalized stationary incompressible flows. Comput. Methods Appl. Mech. Engrg.,190:2681-2706, 2001.
[29]M. R. Correa and A. F. D. Loula. Unconditonally stable mixed finite element methods for darcy flow. Comput. Methods Appl. Mech. Engrg.,197:1525-1540,2008.
[30]G. Danisch and G. Starke. First-order system least-squares for darcy-stokes flow. SIAM J. Number. Anal,45(2):731-745,2007.
[31]C. N. Dawson, T. F. Russell, and M. F. Wheeler. Some improved error estimates for the modified method of characteristics. SIAM J. Numer. Anal., 126:1487-1552,1989.
[32]T. Dupont and L. R. Scott. Polynomial approximation of functions in sobolev spaces. Math. Comput.,34:441-463,1980.
[33]L. Durlofsky and J. F. Brady. Analysis of the brinkman equation as a model for flow in porous media. Phys. Fluids,30:3329-3341,1987.
[34]R. E. Ewing and T. F. Russell. Efficient time-stepping methods for miscible displacement problems in porous media. SIAM J. Numer. Anal.,19:1-67, 1982.
[35]R. E. Ewing, T. F. Russell, and M. F. Wheeler. Simulation of miscible dis-placement using mixed finite elements and a modified method of characteris-tics. SPE 12241, pages 71-81,1983.
[36]R. E. Ewing, T. F. Russell, and M. F. Wheeler. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Engrg.,47:73-92,1984.
[37]R. E. Ewing and H. Wang. An optimal-order estimate for eulerian-lagrangian localized adjoint methods for variable-coefficient advection-reaction problems. SIAM J. Numer. Anal,33:318-348,1996.
[38]R. E. Ewing and H. Wang. A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Math.,128:423-445,2001.
[39]R. E. Ewing, Y. Yuan, and G. Li. Time stepping along characteristics for a mixed finite-element approximation for compressible flow of contamination from nuclear waste in porous media. SIAM J. Numer. Anal.,26:1513-1524, 1989.
[40]J. Feng, P. Ganatos, and S. Weinbaum. Motion of a sphere near planar confining boundaries in a brinkman medium. J. Fluid Mech.,375:265-296, 1998.
[41]B. A. Finlayson. Numerical Methods for Problems with Moving Fronts. Ravenna Park Publishing, Seattle,1992.
[42]M. Fortin. Old and new finite elements for incompressible flows. Int. J. Numer. Methods Fluids,1:347-364,1981.
[43]A. O. Garder, D. W. Peaceman, and A. L. Pozzi. Numerical calculations of multidimensional miscible displacement by mehtod of characteristics. Soc. Pet. Eng. J.,4:26-36,1964.
[44]D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York,1983.
[45]V. Girault and P. A. Raviart. Finite Element Approximations of the Navier-Stokes Equations. Springer-Verlag,1979.
[46]H. Han. An economical finite element scheme for navier-stokes equations. J. Comput. Math.,5:135-143,1987.
[47]H. Han and X. Wu. A new mixed finite element formulation and the mac method for the stokes equations. SIAM J. Numer. Anal.,35:560-571,1998.
[48]H. Han and M. Yan. A mixed finite element method on a staggered mesh for navier-stokes equations. J. Compu. Math.,26:815-824,2008.
[49]P. Hansbo and M. Juntunen. Weakly imposed dirichlet boundary conditions for the brinkman model of porous media flow. Appl. Numer. Math.,59:1274-1289,2009.
[50]T. J. R. Hughes, A. Masud, and J. Wan. A stabilized mixed discontinu-ous galerkin method for darcy flow. Comput. Methods Appl. Mech. Engrg., 195:3347-3381,2006.
[51]J. Douglas Jr., R. E. Ewing, and M. F. Wheeler. The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Model. Math. Anal. Numer.,17:17-33,1983.
[52]J. Douglas Jr., R. E. Ewing, and M. F. Wheeler. A time-discretization pro-cedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Model. Math. Anal. Numer.,17:249-265,1983.
[53]J. Douglas Jr. and J. E. Roberts. Numerical methods for a model for com-pressible miscible displacement in porous media. Math. Comp.,41:441-459, 1983.
[54]J. Douglas Jr. and T. F. Russell. Numerical methods for convection-dominated diffusion problems based on combining the method of character-istics with finite element or finite difference procedures. SIAM J. Numer. Anal.,19:871-885,1982.
[55]O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva. Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, RI,1968.
[56]W. J. Layton, F. Schieweck, and I. Yotov. Coupling fluid with porous media flow. SIAM J. Numer. Anal.,40:2195-2218,2003.
[57]T Levy. Loi de darcy ou loi de brinkman?. C.R. Acad. Sci. Paris Serie II Mec. Phys. Chim.Sci.Univers Sci.,292:871-874,1981.
[58]J. L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York,1972.
[59]K. A. Mardal, X-C. Tai. and R. Winther. A robust finite element method for dacry-stoke flow. SIAM J. Numer. Anal.,40:1605-1631,2002.
[60]A. Masud and T. J. R. Hughes. A stabilized finite element method for darcy flow. Comput. Methods Appl. Mech. Engrg.,191:4341-4370,2002.
[61]T. K. Nilssen, X-C. Tai, and R. Winther. A robust nonconforming h2-element. Math. Comp.,70:489-505,2000.
[62]D. W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Company,1977.
[63]A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer-Ver lag, Berlin,1994.
[64]R. Rannacher and R. Scott. Some optimal error estimates for piecewise linear finite element approximations. Math. Comp.,28:437-445,1982.
[65]P. A. Raviart and J. M. Thomas. A mixed finite element method for second order elliptic problem. Mathematical Aspects of the FEM, Lecture Notes in Mathematics,606:292-315,1977.
[66]B. Riviere and I. Yotov. Locally conservative coupling of stokes and darcy flows. SIAM J. Numer. Anal.,42:1959-1977,2005.
[67]H. Rui and M. Tabata. A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math.,92:161-177,2002.
[68]H. Rui and M. Tabata. A mass-conservative finite element scheme for convection-diffusion problems. J. Sci. Comput.,43:416-432,2010.
[69]H. Rui and R. Zhang. A unified stabilized mixed finite element method for coupling stokes and darcy flows. Comput. Methods Appl. Mesh. Engrg., 198:2692-2699,2009.
[70]T. F. Russell. Finite elements with characteristics for two-component incom-pressible miscible displacement. SPE 10500, in:Proc.6th SPE Symposium on Reservoir Simulation, pages 123-135,1982.
[71]T. F. Russell. Time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal,22:970-1013,1985.
[72]P. G. Saffman. On the boundary condition at the surface of a porous media. Stud. Appl. Math.,50:93-101,1971.
[73]L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput.,54:483-493,1990.
[74]E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton,1970.
[75]R. Stenberg. On some techniques for aproximating boundary conditions in the finite element method. J. Comput. Appl. Math.,63(1-3):139-148,1995.
[76]V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin,1997.
[77]H. Wang. An optimal-order error estimate for an ellam scheme for two-dimensional linear advection-diffusion equations. SIAM J. Numer. Anal., 37:1338-1368,2000.
[78]H. Wang, R. E. Ewing, and T. F. Russell. Eulerian-lagrangian localized methods for convection-diffusion equations and their convergence analysis. SIAM J. Numer. Anal,15:405-459,1995.
[79]H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, and G. Qin. An approxi-mation to miscible fluid flows in porous media with point sources and sinks by an eulerian-lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comp.,22:561-581,2000.
[80]H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, and G. Qin. An ellam-mfem solution technique for compressible fluid flows in porous media with point sources and sinks.J. Comput. Math.,159:344-376,2000.
[81]M. F. Wheeler. A priori l2 error estimates for galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal.,10:723-759, 1973.
[82]X. P. Xie, J. C. Xu. and G. R. Xue. Uniformly-stable finite element methods for darcy-stokes-brinlman models. J. Comput. Math.,26:437-455,2008.
[83]X. J. Xu and S. Y. Zhang. A new divergence-free interpolation operator with applications to the darcy-stokes-brinkman equaion. SIAM J. Sci. Comp., 32:855-874,2010.
[84]K. Yosida. Functional Analysis. Springer-Verlag, New York, sixth edition, 1980.
[85]Y. Yuan. Time stepping along characteristics for the finite element approxi-mation of compressible miscible dispacement in porous media. Mathematical Numerica Sinica(in Chinese),14:385-400,1992.
[2]P. Angot. Analysis of singular perturbation on the brinkman problem for fictious domain models of viscous flows. Math. Methods Appl. Sci.,22:1395-1412,1999.
[3]T. Arbogast and D. D. Brunson. A computational method for approximating a darcy-stokes system governing a vuggy porous medium. Comput. Geosci., 11:207-218,2007.
[4]T. Arbogast and H. L. Lehr. Homogenization of a darcy-stokes system mod-eling vuggy porous media. Comput. Geosci.,10(3):291-302,2006.
[5]T. Arbogast and M. F. Wheeler. A characteristics-mixed finite element method for advection-dominated transport problems. SIAM J. Numer. Anal., 32:404-424,1995.
[6]T. Arbogast and M. F. Wheeler. A family of rectangular mixed element with a continuous flux for second order elliptic problems. SIAM J. Numer. Anal., 42:1914-1931,2005.
[7]I. Babuska. Error bounds for finite element methods. Numer. Math.,16:322-333,1971.
[8]L. Badea, M. Discacciati, and A.Quarteroni. Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Mi-lan,2006.
[9]S. Badia and R. Codina. Unified stabilized finite element formulations for the stokes and the darcy problems. SIAM J. Numer. Anal.,47(3):1971-2000, 2009.
[10]J. W. Barrett and K. W. Morton. Approximate symmetrization and petrov-galerkin methods for diffusion-convection problems. Comput. Methods Appl. Mech. Eng.,45:97-122,1984.
[11]J. Bear. Dynamics of Fluids in Porous Media. Elsevier, New York,1972.
[12]G. S. Beavers and D. D. Joseph. Boundary conditions at a naturally imper-meable wall. J. Fluid Mech.,30:197-207,1967.
[13]J. M. Boland and R. A. Nicolaides. On the stability of bilinear-constant velocity-pressure finite elements. Numer. Math.,44:219-222,1984.
[14]S. C. Brenner. Korn's inequalities for piecewise hl vector fields. Math. Com-put.,73:1067-1087,2003.
[15]S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York,1994.
[16]F. Brezzi. On the existence, uniqueness and approximation of saddle point problems arising from lagrangian multipliers. RAIRO Model. Math. Anal. Numer.,8:129-151,1974.
[17]F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, 1991.
[18]F. Brezzi. J. Douglas Jr., R. Duran, and M. Fortin. Mixed finite elements for second order elliptic problems in three variables. Numer. Math.,51:237-250, 1987.
[19]F. Brezzi, J. Douglas Jr., M. Fortin, and L. D. Marini. Efficient rectangular mixed finite elements in two and three space variables. Math. Model. Numer. Anal.,21:581-604,1987.
[20]F. Brezzi, J. Douglas Jr., and L. D. Marini. Two families of mixed elements for second order ellptic problems. Numer. Math.,88:217-235,1985.
[21]E. Burman and P. Hansbo. Stabilized crouzeix-raviart element for the darcy-stokes problem. Numer Methods Partial Differ. Eq.,21:986-997,2005.
[22]M. A. Celia, T. F. Russell, I. Herrera, and R. E. Ewing. An eulerian-lagrangian localized adjoint method for the advection-diffusion equation. Ad-vances in Water Resources,13:187-206,1990.
[23]Z. Chen and R. E. Ewing. Fully discrete finite element analysis of multiphase flow in groundwater hydrology. SIAM J. Numer. Anal.,34:2228-253,1977.
[24]A. Cheng and G. Wang. The modified method of characteristics for com-pressible flow in porous media. Lecture Notes in Physics,552:69-79,2000.
[25]S. H. Chou and Q. Li. Mixed finite element methods for compressible miscible displacement in porous media. Mathematics of Computation,57(196):507-527,1991.
[26]P. G. Ciarlet. The Finite Element Method for Elliptic Problems. Amsterdam, North-Holland,1978.
[27]P. G. Ciarlet and P. A. Raviart. General lagrange and hermite interpolation in rn with applications to finite element methods. Arch. Rat. Mech. Anal., 46:177-199,1972.
[28]R. Codina. A stabilized finite element method for generalized stationary incompressible flows. Comput. Methods Appl. Mech. Engrg.,190:2681-2706, 2001.
[29]M. R. Correa and A. F. D. Loula. Unconditonally stable mixed finite element methods for darcy flow. Comput. Methods Appl. Mech. Engrg.,197:1525-1540,2008.
[30]G. Danisch and G. Starke. First-order system least-squares for darcy-stokes flow. SIAM J. Number. Anal,45(2):731-745,2007.
[31]C. N. Dawson, T. F. Russell, and M. F. Wheeler. Some improved error estimates for the modified method of characteristics. SIAM J. Numer. Anal., 126:1487-1552,1989.
[32]T. Dupont and L. R. Scott. Polynomial approximation of functions in sobolev spaces. Math. Comput.,34:441-463,1980.
[33]L. Durlofsky and J. F. Brady. Analysis of the brinkman equation as a model for flow in porous media. Phys. Fluids,30:3329-3341,1987.
[34]R. E. Ewing and T. F. Russell. Efficient time-stepping methods for miscible displacement problems in porous media. SIAM J. Numer. Anal.,19:1-67, 1982.
[35]R. E. Ewing, T. F. Russell, and M. F. Wheeler. Simulation of miscible dis-placement using mixed finite elements and a modified method of characteris-tics. SPE 12241, pages 71-81,1983.
[36]R. E. Ewing, T. F. Russell, and M. F. Wheeler. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Engrg.,47:73-92,1984.
[37]R. E. Ewing and H. Wang. An optimal-order estimate for eulerian-lagrangian localized adjoint methods for variable-coefficient advection-reaction problems. SIAM J. Numer. Anal,33:318-348,1996.
[38]R. E. Ewing and H. Wang. A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Math.,128:423-445,2001.
[39]R. E. Ewing, Y. Yuan, and G. Li. Time stepping along characteristics for a mixed finite-element approximation for compressible flow of contamination from nuclear waste in porous media. SIAM J. Numer. Anal.,26:1513-1524, 1989.
[40]J. Feng, P. Ganatos, and S. Weinbaum. Motion of a sphere near planar confining boundaries in a brinkman medium. J. Fluid Mech.,375:265-296, 1998.
[41]B. A. Finlayson. Numerical Methods for Problems with Moving Fronts. Ravenna Park Publishing, Seattle,1992.
[42]M. Fortin. Old and new finite elements for incompressible flows. Int. J. Numer. Methods Fluids,1:347-364,1981.
[43]A. O. Garder, D. W. Peaceman, and A. L. Pozzi. Numerical calculations of multidimensional miscible displacement by mehtod of characteristics. Soc. Pet. Eng. J.,4:26-36,1964.
[44]D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York,1983.
[45]V. Girault and P. A. Raviart. Finite Element Approximations of the Navier-Stokes Equations. Springer-Verlag,1979.
[46]H. Han. An economical finite element scheme for navier-stokes equations. J. Comput. Math.,5:135-143,1987.
[47]H. Han and X. Wu. A new mixed finite element formulation and the mac method for the stokes equations. SIAM J. Numer. Anal.,35:560-571,1998.
[48]H. Han and M. Yan. A mixed finite element method on a staggered mesh for navier-stokes equations. J. Compu. Math.,26:815-824,2008.
[49]P. Hansbo and M. Juntunen. Weakly imposed dirichlet boundary conditions for the brinkman model of porous media flow. Appl. Numer. Math.,59:1274-1289,2009.
[50]T. J. R. Hughes, A. Masud, and J. Wan. A stabilized mixed discontinu-ous galerkin method for darcy flow. Comput. Methods Appl. Mech. Engrg., 195:3347-3381,2006.
[51]J. Douglas Jr., R. E. Ewing, and M. F. Wheeler. The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Model. Math. Anal. Numer.,17:17-33,1983.
[52]J. Douglas Jr., R. E. Ewing, and M. F. Wheeler. A time-discretization pro-cedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Model. Math. Anal. Numer.,17:249-265,1983.
[53]J. Douglas Jr. and J. E. Roberts. Numerical methods for a model for com-pressible miscible displacement in porous media. Math. Comp.,41:441-459, 1983.
[54]J. Douglas Jr. and T. F. Russell. Numerical methods for convection-dominated diffusion problems based on combining the method of character-istics with finite element or finite difference procedures. SIAM J. Numer. Anal.,19:871-885,1982.
[55]O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva. Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, RI,1968.
[56]W. J. Layton, F. Schieweck, and I. Yotov. Coupling fluid with porous media flow. SIAM J. Numer. Anal.,40:2195-2218,2003.
[57]T Levy. Loi de darcy ou loi de brinkman?. C.R. Acad. Sci. Paris Serie II Mec. Phys. Chim.Sci.Univers Sci.,292:871-874,1981.
[58]J. L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York,1972.
[59]K. A. Mardal, X-C. Tai. and R. Winther. A robust finite element method for dacry-stoke flow. SIAM J. Numer. Anal.,40:1605-1631,2002.
[60]A. Masud and T. J. R. Hughes. A stabilized finite element method for darcy flow. Comput. Methods Appl. Mech. Engrg.,191:4341-4370,2002.
[61]T. K. Nilssen, X-C. Tai, and R. Winther. A robust nonconforming h2-element. Math. Comp.,70:489-505,2000.
[62]D. W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Company,1977.
[63]A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer-Ver lag, Berlin,1994.
[64]R. Rannacher and R. Scott. Some optimal error estimates for piecewise linear finite element approximations. Math. Comp.,28:437-445,1982.
[65]P. A. Raviart and J. M. Thomas. A mixed finite element method for second order elliptic problem. Mathematical Aspects of the FEM, Lecture Notes in Mathematics,606:292-315,1977.
[66]B. Riviere and I. Yotov. Locally conservative coupling of stokes and darcy flows. SIAM J. Numer. Anal.,42:1959-1977,2005.
[67]H. Rui and M. Tabata. A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math.,92:161-177,2002.
[68]H. Rui and M. Tabata. A mass-conservative finite element scheme for convection-diffusion problems. J. Sci. Comput.,43:416-432,2010.
[69]H. Rui and R. Zhang. A unified stabilized mixed finite element method for coupling stokes and darcy flows. Comput. Methods Appl. Mesh. Engrg., 198:2692-2699,2009.
[70]T. F. Russell. Finite elements with characteristics for two-component incom-pressible miscible displacement. SPE 10500, in:Proc.6th SPE Symposium on Reservoir Simulation, pages 123-135,1982.
[71]T. F. Russell. Time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal,22:970-1013,1985.
[72]P. G. Saffman. On the boundary condition at the surface of a porous media. Stud. Appl. Math.,50:93-101,1971.
[73]L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput.,54:483-493,1990.
[74]E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton,1970.
[75]R. Stenberg. On some techniques for aproximating boundary conditions in the finite element method. J. Comput. Appl. Math.,63(1-3):139-148,1995.
[76]V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin,1997.
[77]H. Wang. An optimal-order error estimate for an ellam scheme for two-dimensional linear advection-diffusion equations. SIAM J. Numer. Anal., 37:1338-1368,2000.
[78]H. Wang, R. E. Ewing, and T. F. Russell. Eulerian-lagrangian localized methods for convection-diffusion equations and their convergence analysis. SIAM J. Numer. Anal,15:405-459,1995.
[79]H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, and G. Qin. An approxi-mation to miscible fluid flows in porous media with point sources and sinks by an eulerian-lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comp.,22:561-581,2000.
[80]H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, and G. Qin. An ellam-mfem solution technique for compressible fluid flows in porous media with point sources and sinks.J. Comput. Math.,159:344-376,2000.
[81]M. F. Wheeler. A priori l2 error estimates for galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal.,10:723-759, 1973.
[82]X. P. Xie, J. C. Xu. and G. R. Xue. Uniformly-stable finite element methods for darcy-stokes-brinlman models. J. Comput. Math.,26:437-455,2008.
[83]X. J. Xu and S. Y. Zhang. A new divergence-free interpolation operator with applications to the darcy-stokes-brinkman equaion. SIAM J. Sci. Comp., 32:855-874,2010.
[84]K. Yosida. Functional Analysis. Springer-Verlag, New York, sixth edition, 1980.
[85]Y. Yuan. Time stepping along characteristics for the finite element approxi-mation of compressible miscible dispacement in porous media. Mathematical Numerica Sinica(in Chinese),14:385-400,1992.