对流扩散方程的特征有限元方法
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摘要
对流扩散方程作为偏微分方程一个很重要的分支,在众多领域都有着广泛的应用,如流体力学,气体动力学等.由于对流扩散方程很难通过解析的方法得到解析解,所以通过各种数值方法来求解对流扩散方程在数值分析中占有很重要的地位.在对流扩散方程中,若扩散项在物理过程中起主导作用.则用标准有限差分方法以及有限元方法求解就可以得到很好的数值结果.但是,若对流项是占主导地位,即对流的影响远大于扩散的影响,则会给数值求解带来很多困难,如数值震荡,数值的过度扩散,或者是二者皆有.在处理对流占优的对流扩散方程的数值方法中,很重要的一类方法就是特征线法.这一方法考虑沿特征线(流动方向)作离散,利用对流扩散问题的物理学特征,对于处理具有双曲性质的对流占优扩散问题具有无可比拟的优越性.它不仅可以从本质上减少非物理震荡和过多的数值弥散,而且对时间步长没有稳定性限制条件,并且沿特征方向的导数值远比沿时间方向相应的导数值小.
对于特征线法,前人已经有了很多数学上的分析以及实际应用上的研究工作.上世纪六十年代人们构造了向前追踪的特征线方法(MOC)[32]，直到近年来,人们还在不断的改进这一方法,并将它们广泛的应用到许多实际问题.这类方法对于相对简单的问题比较容易实现，但是沿特征线向前追踪扭曲了原有空间网格、给计算带来了很多不便.1982年Douglas和Russell在[24]中提出了沿特征线向后追踪的修正特征线法(MMOC),克服了原有特征线法的缺点.这种方法也得到了广泛的应用.如[4],[46]等将这类特征线法与混合元等方法结合来处理多孔介质混溶驱动问题.但是(MMOC)不能满足质量守恒,不久Douglas等又构造了校正对流项的特征线方法(MMOCAA)[22].[23]该方法可以满足整体质量守恒.在[51]中.芮洪兴和Tabata也提出了一种新型的特征有限元方法，这种方法保持了对流扩散过程质量守恒的特性.在1990年Celia等人提出了欧拉-拉格朗日局部共轭法(ELLAM)[14]这种方法不仅能保持质量守恒性,而且能够方便的处理边界条件,但是缺点是计算所得到的积分有一定的困难.芮洪兴和Tabata在2002年还提出了一种新的特征线方法[50],这种计算格式在时间步长上具有二阶精度,并且是对称的和无条件稳定的.
多孔介质中流体流动的数学物理模型在数学上表现为依赖于时间的强耦合的非线性偏微分方程组,该方程组结构复杂,只有在特殊的情形下才有解析解的表达式.故对其进行数值模拟也是计算数学的一项非常重要的课题.多孔介质流的模型通常由两部分组成,一是质量守恒,主要体现在物质的平衡,如注产体积、质量的平衡,这可以通过对流扩散方程来描述,另一个就是动量守恒,主要用各种速度与压力的关系式来描述,如作为经验公式引入的达西定律等.假设流体不可压缩,那么就可以简化为由强对流扩散方程和达西方程耦合的描述多孔介质混溶驱动问题的经典模型.对于这一经典模型,前人已经做了很多经典的研究分析工作,也得到了广泛的应用Russell在[53]中对浓度方程运用特征有限元方法,对压力方程运用标准Galerkin方法分析了这一模型.用特征线方法求解浓度方程可以使方程对称化,增强了稳定性,减少时间截断误差.可以使用较大的时间步长，对压力方程用标准Galerkin方法只能求出压力,不能直接求出速度Ewing和Russell在[25][28]中运用MMOC方法求解对流扩散方程.用混合元方法求解压力和速度方程,这种方法可以同时求出压力和速度,不需要通过对压力求微分来得到速度的近似值.从而减低了速度误差.
本文的另一个研究重点是变网格方法.用有限元方法求解依赖于时间的问题.通常的做法是在空间采用有限元方法,而在时间方向上采用有限差分格式.以往人们提出的算法大都是限制在空间区域的固定有限元网格上.然而.在许多实际计算问题中,往往需要在不同的时间层采用不同的有限元空间例如.火焰的传播以及油水前沿面问题等.又如抛物型方程的初始值,若光滑性较差,则在开始的一段时间内.由于真解的光滑性较差,应采用较低阶插值函数和较密的网格,经过一段时间.真解的光滑性变好,则可以用较高阶的插值函数和较粗的网格.因此.许多数学家和工程师都把目光放在采用动态有限元空间这一方法上.而且也提出了许多动态有限元方法.梁国平在[39]给出了一般抛物问题的变网格有限元方法,这种方法的主要思路是根据需要对不同的时间层采用不同的空间网格,把上一时间层的近似解L2投影到当前时间层然后作为初始值；而后又基于坐标变换的思想,提出了适用于任何空间维数的一般抛物方程以及任何网格的非柱形区域的变网格有限元离散格式[41]；杨道奇[61]对抛物问题提出了变网格混合元方法,随后又将该方法推广到多孔介质可混溶驱动问题[62]；袁益让教授[65]讨论了非线性对流扩散问题的变网格方法.
本文围绕着对流扩散方程的特征线法而展开,主要分析了二阶特征线的变网格方法,二阶特征线法与Galerkin有限元法相结合处理不可压多孔介质混溶驱动问题,最后给出了守恒特征有限元法的一种最优估计.
本文的组织结构如下：
在第一章中,我们建立本文要讨论的数学模型,通过多孔介质混溶驱动问题的物理背景,根据质量守恒性,推导描述多孔介质中流体浓度的对流占优的对流扩散方程,及其与描述流体速度和压力关系的达西方程组成的耦合非线性偏微分方程.接着介绍了本文中会用到的sobolev空间及其范数,并给出了后面章节分析中会经常用到几个引理.
在第二章中,我们把二阶特征线法与变网格法相结合给出了对流占优的对流扩散方程的数值形式,并做了相应的误差分析.证明近似算法在时间变量方向是二阶的，而当网格变动次数M满足一定的条件时,在空间变量方向上虽然不是最优的但依然有较好的收敛速度.当Mh有界时,L2模的收敛速度为hk阶的.
在第三章中，我们进行了不可压多孔介质混溶驱动问题的数值分析，我们运用二阶特征有限元的方法处理浓度方程,而对压力方程采用标准Galerkin方法进行处理.我们给出了问题的假设以及二阶特征Galerkin方法的离散形式并给出了相应的L2模误差估计.证明了算法在时间变量方向是二阶的.在空间变量方向上虽不是最优的但也有较好的收敛速度.
在第四章中,我们对芮洪兴及Tabata提出的新型守恒特征有限元方法,在一维均匀剖分的情况下,利用超收敛估计,给出了其最优估计并用数值算例进一步验证了结论的正确性.
Convection-diffusion equations are an important class of partial differ-ential equations that arise in many scientific fields including fluid mechanics, gas dynamics and so on. Since these equations normally have no closed form analytical solutions, it is very important to have accurate numerical approx-imations. When diffusion dominates the physical process, standard finite difference methods and finite element methods work well in solving these equations. However, when convection is the dominant process, these equa-tions present many numerical difficulties including non-physical oscillations, excessive numerical diffusion which smears out sharp fronts, or a combination of both. In the framework of numerical solutions of convection dominated problems. a possible strategy is provided by the method of characteristics for time discretization. This approach is based on the discretization of the material derivative, and it takes advantage of the physics characteristics of the convoction-diffusion problems, has incomparable superiority for dealing with hyperbolic nature of convection-dominated diffusion problems. It, can not only substantially reduces the non-physical oscillations, excessive numer-ical diffusion, but also has no stability constraints on the time step, and the derivative value of the solution along the characteristic direction is smaller than the value along the time direction.
Many authors have mathematically analyzed and applied this method to different problems. In the sixties of the last century, people construct a for-ward tracking method of characteristic (MOC)[32], until recent years, people continue to improve this method, and apply it to many practical problems. This type of method is relatively easy to implemented to simple problems, but the tracking along the characteristic line destroys the original space grid. and brought a lot of inconvenience to the calculation. In1982, Douglas and Russell [24]proposed a back tracking method of characteristic (modified method of characteristics MMOC), which overcomes the shortcomings of the original method of characteristics. This method has also been widely used, for example [46],[4] which combined this kind of method with mixed element method to handle the miscible displacement in porous media. But the back tracking method can not satisfy the mass conservation. Soon after that, Douglas and others constructed the modified method of characteristics with adjusted advection (MMOC A A)[22],[23] which can satisfy the overall mass conservation. In [51], professor Rui Hongxing and Tabata also constructed a new method of characteristics which also maintains the mass conservation of the convection diffusion process. In1990, Celia and others proposed the Euler-Lagrange localized adjoint method (ELLAM)[14], which not only can maintain the mass conservation, but also facilitates the treatment of boundary conditions, but the drawback is that the calculations of the inte-gral has certain difficulty. In [50], professor Rui Hongxing and Tabata first introduced a second order characteristic finite element method. This numer-ical scheme is of second order accuracy in time increment, symmetric and unconditionally stable.
Mathematical and physical model for the fluid flow and transport pro-cesses in porous media leads to a highly coupled time-dependent nonlinear partial differential equations. Because of the complex structure of the partial differential equations, analytical solution only exists in some special cases. Therefore, large-scale, high-accuracy simulation for the model is a urgent necessity in science and engineering. In sight of the complexity of the model for fluid flow in porous media, mass conservation results in the mass balance of the injection and production, which can be described by the convection-diffusion equations, and the momentum conservation is often approximated by some simplified velocity and pressure relation, experimental formulas for example, the Darcy's law. Assuming that the fluid is incompressible, the convection dominated diffusion equation combined with Darcy's law is the widely used classical model. A lot of research and analysis have been done for the model, and also get a wide range of applications. Russell [53] has ana-lyzed the concentration equation by a combination of a Galerkin method and the method of characteristics and the press equation by a standard Galerkin procedure. Using the method of characteristics for solving the concentra-tion equation can make the equation symmetric, enhance stability, reduce the time truncation error, and the larger time step can be used. Pressure equation using the standard Galerkin method can only determine the pres-sure, but the speed can not be obtained directly. Ewing and Russell [28][25] have analyzed the coupled process by mixed finite elements and a modi-fied method of characteristics. This method can be obtained simultaneously by the pressure and velocity, without going through the differentiating the pressure to obtain an approximation of the velocity, thus reducing the speed error.
Another research focus of this article is the moving grid method. Us-ing the finite element method for solving the time-dependent problems (such as parabolic), the usually effective practice in in the region of space using the finite element method, and in the time direction using a finite difference scheme. A fixed set of finite element mesh have been proposed in the past algorithms which is limited to the region of space Grid. However, in many practical computing problems, we often need in different layers of time using different finite element spaces. For example, the flame pass Broadcast. and oil-water frontier. Therefore, many mathematicians and engineers focus on the dynamic finite element spare method, and many dynamic finite element methods have been proposed. Liang Guoping [39]proposed a moving grid finite element method for general parabolic problems. The main idea of the method is that a different number of finite element spaces is adopted at differ-ent time level, and the approximate solution at the current time is projected in the L2-norm onto the next time finite clement space and make it as an initial value. By developing the idea of coordinate transformation, he then presents a full-discretization moving finite element scheme[41], which can ap- ply to parabolic equations in any dimension with variable domain and under certain continuity assumptions about moving grid the optimal convergence rate is preserved. Yang Daoqi [61]proposed a mixed finite element method with moving grid for the parabolic problems, and extended to the porous medium miscible displacement [62]. Professor Yuan Yirang [65] discussed the moving grid method for nonlinear convection-diffusion problems.
This article is based on the finite element method of characteristics for the convection-diffusion equation. We mainly discussed the second or-der characteristics finite element method with moving grid, the second-order characteristics finite element method combined with the Galerkin finite ele-ment method in incompressible miscible displacement in porous media, and finally we gave an optimal-order error estimate for the mass-conservative characteristic finite clement scheme.
In Chapter1, we give the mathematical model which is discussed in this paper. According to the physical background of the miscible displace-ment in porous media, from the nature of mass conservation, we derive the convection-diffusion equation describing the concentration of fluids in porous media, and combining with Darcy equations describing fluid velocity and pressure relationship, we derive the coupled nonlinear partial differential equations. Then we introduce the sobolev spaces and its norms and give several lemmas frequently used in the later chapters.
In Chapter2, we disease the second order characteristics finite element method with moving grid. The approximation algorithm is of second order accuracy in time increment, and when the number of the moved grids M satisfies certain conditions, though it is not optimal accuracy in space incre-ment, but still have good convergence rate. When Mh is bounded, it has a convergence rate of hk in L2-norm.
In Chapter3, we develop the numerical analysis of incompressible mis-cible displacement in porous media, we use the second-order characteristics finite element method to treat the concentration equation and the Galerkin finite element method in the press equation. We list our assumptions on the problem, define the second order characteristic-Galerkin method and give the error estimates in L2-norm. Finally, we prove that the approximation algorithm is of second order accuracy in time increment, and though it is not optimal accuracy in space increment, but still have good convergence rate.
In Chapter3, an optimal-order error estimate is obtained for the model problem in one space dimension. The convergence rate is O(h2+Δt) in the l∞(L2) norm for the linear finite element. Some numerical examples are given to confirm the conclusion.
对于特征线法,前人已经有了很多数学上的分析以及实际应用上的研究工作.上世纪六十年代人们构造了向前追踪的特征线方法(MOC)[32]，直到近年来,人们还在不断的改进这一方法,并将它们广泛的应用到许多实际问题.这类方法对于相对简单的问题比较容易实现，但是沿特征线向前追踪扭曲了原有空间网格、给计算带来了很多不便.1982年Douglas和Russell在[24]中提出了沿特征线向后追踪的修正特征线法(MMOC),克服了原有特征线法的缺点.这种方法也得到了广泛的应用.如[4],[46]等将这类特征线法与混合元等方法结合来处理多孔介质混溶驱动问题.但是(MMOC)不能满足质量守恒,不久Douglas等又构造了校正对流项的特征线方法(MMOCAA)[22].[23]该方法可以满足整体质量守恒.在[51]中.芮洪兴和Tabata也提出了一种新型的特征有限元方法，这种方法保持了对流扩散过程质量守恒的特性.在1990年Celia等人提出了欧拉-拉格朗日局部共轭法(ELLAM)[14]这种方法不仅能保持质量守恒性,而且能够方便的处理边界条件,但是缺点是计算所得到的积分有一定的困难.芮洪兴和Tabata在2002年还提出了一种新的特征线方法[50],这种计算格式在时间步长上具有二阶精度,并且是对称的和无条件稳定的.
多孔介质中流体流动的数学物理模型在数学上表现为依赖于时间的强耦合的非线性偏微分方程组,该方程组结构复杂,只有在特殊的情形下才有解析解的表达式.故对其进行数值模拟也是计算数学的一项非常重要的课题.多孔介质流的模型通常由两部分组成,一是质量守恒,主要体现在物质的平衡,如注产体积、质量的平衡,这可以通过对流扩散方程来描述,另一个就是动量守恒,主要用各种速度与压力的关系式来描述,如作为经验公式引入的达西定律等.假设流体不可压缩,那么就可以简化为由强对流扩散方程和达西方程耦合的描述多孔介质混溶驱动问题的经典模型.对于这一经典模型,前人已经做了很多经典的研究分析工作,也得到了广泛的应用Russell在[53]中对浓度方程运用特征有限元方法,对压力方程运用标准Galerkin方法分析了这一模型.用特征线方法求解浓度方程可以使方程对称化,增强了稳定性,减少时间截断误差.可以使用较大的时间步长，对压力方程用标准Galerkin方法只能求出压力,不能直接求出速度Ewing和Russell在[25][28]中运用MMOC方法求解对流扩散方程.用混合元方法求解压力和速度方程,这种方法可以同时求出压力和速度,不需要通过对压力求微分来得到速度的近似值.从而减低了速度误差.
本文的另一个研究重点是变网格方法.用有限元方法求解依赖于时间的问题.通常的做法是在空间采用有限元方法,而在时间方向上采用有限差分格式.以往人们提出的算法大都是限制在空间区域的固定有限元网格上.然而.在许多实际计算问题中,往往需要在不同的时间层采用不同的有限元空间例如.火焰的传播以及油水前沿面问题等.又如抛物型方程的初始值,若光滑性较差,则在开始的一段时间内.由于真解的光滑性较差,应采用较低阶插值函数和较密的网格,经过一段时间.真解的光滑性变好,则可以用较高阶的插值函数和较粗的网格.因此.许多数学家和工程师都把目光放在采用动态有限元空间这一方法上.而且也提出了许多动态有限元方法.梁国平在[39]给出了一般抛物问题的变网格有限元方法,这种方法的主要思路是根据需要对不同的时间层采用不同的空间网格,把上一时间层的近似解L2投影到当前时间层然后作为初始值；而后又基于坐标变换的思想,提出了适用于任何空间维数的一般抛物方程以及任何网格的非柱形区域的变网格有限元离散格式[41]；杨道奇[61]对抛物问题提出了变网格混合元方法,随后又将该方法推广到多孔介质可混溶驱动问题[62]；袁益让教授[65]讨论了非线性对流扩散问题的变网格方法.
本文围绕着对流扩散方程的特征线法而展开,主要分析了二阶特征线的变网格方法,二阶特征线法与Galerkin有限元法相结合处理不可压多孔介质混溶驱动问题,最后给出了守恒特征有限元法的一种最优估计.
本文的组织结构如下：
在第一章中,我们建立本文要讨论的数学模型,通过多孔介质混溶驱动问题的物理背景,根据质量守恒性,推导描述多孔介质中流体浓度的对流占优的对流扩散方程,及其与描述流体速度和压力关系的达西方程组成的耦合非线性偏微分方程.接着介绍了本文中会用到的sobolev空间及其范数,并给出了后面章节分析中会经常用到几个引理.
在第二章中,我们把二阶特征线法与变网格法相结合给出了对流占优的对流扩散方程的数值形式,并做了相应的误差分析.证明近似算法在时间变量方向是二阶的，而当网格变动次数M满足一定的条件时,在空间变量方向上虽然不是最优的但依然有较好的收敛速度.当Mh有界时,L2模的收敛速度为hk阶的.
在第三章中，我们进行了不可压多孔介质混溶驱动问题的数值分析，我们运用二阶特征有限元的方法处理浓度方程,而对压力方程采用标准Galerkin方法进行处理.我们给出了问题的假设以及二阶特征Galerkin方法的离散形式并给出了相应的L2模误差估计.证明了算法在时间变量方向是二阶的.在空间变量方向上虽不是最优的但也有较好的收敛速度.
在第四章中,我们对芮洪兴及Tabata提出的新型守恒特征有限元方法,在一维均匀剖分的情况下,利用超收敛估计,给出了其最优估计并用数值算例进一步验证了结论的正确性.
Convection-diffusion equations are an important class of partial differ-ential equations that arise in many scientific fields including fluid mechanics, gas dynamics and so on. Since these equations normally have no closed form analytical solutions, it is very important to have accurate numerical approx-imations. When diffusion dominates the physical process, standard finite difference methods and finite element methods work well in solving these equations. However, when convection is the dominant process, these equa-tions present many numerical difficulties including non-physical oscillations, excessive numerical diffusion which smears out sharp fronts, or a combination of both. In the framework of numerical solutions of convection dominated problems. a possible strategy is provided by the method of characteristics for time discretization. This approach is based on the discretization of the material derivative, and it takes advantage of the physics characteristics of the convoction-diffusion problems, has incomparable superiority for dealing with hyperbolic nature of convection-dominated diffusion problems. It, can not only substantially reduces the non-physical oscillations, excessive numer-ical diffusion, but also has no stability constraints on the time step, and the derivative value of the solution along the characteristic direction is smaller than the value along the time direction.
Many authors have mathematically analyzed and applied this method to different problems. In the sixties of the last century, people construct a for-ward tracking method of characteristic (MOC)[32], until recent years, people continue to improve this method, and apply it to many practical problems. This type of method is relatively easy to implemented to simple problems, but the tracking along the characteristic line destroys the original space grid. and brought a lot of inconvenience to the calculation. In1982, Douglas and Russell [24]proposed a back tracking method of characteristic (modified method of characteristics MMOC), which overcomes the shortcomings of the original method of characteristics. This method has also been widely used, for example [46],[4] which combined this kind of method with mixed element method to handle the miscible displacement in porous media. But the back tracking method can not satisfy the mass conservation. Soon after that, Douglas and others constructed the modified method of characteristics with adjusted advection (MMOC A A)[22],[23] which can satisfy the overall mass conservation. In [51], professor Rui Hongxing and Tabata also constructed a new method of characteristics which also maintains the mass conservation of the convection diffusion process. In1990, Celia and others proposed the Euler-Lagrange localized adjoint method (ELLAM)[14], which not only can maintain the mass conservation, but also facilitates the treatment of boundary conditions, but the drawback is that the calculations of the inte-gral has certain difficulty. In [50], professor Rui Hongxing and Tabata first introduced a second order characteristic finite element method. This numer-ical scheme is of second order accuracy in time increment, symmetric and unconditionally stable.
Mathematical and physical model for the fluid flow and transport pro-cesses in porous media leads to a highly coupled time-dependent nonlinear partial differential equations. Because of the complex structure of the partial differential equations, analytical solution only exists in some special cases. Therefore, large-scale, high-accuracy simulation for the model is a urgent necessity in science and engineering. In sight of the complexity of the model for fluid flow in porous media, mass conservation results in the mass balance of the injection and production, which can be described by the convection-diffusion equations, and the momentum conservation is often approximated by some simplified velocity and pressure relation, experimental formulas for example, the Darcy's law. Assuming that the fluid is incompressible, the convection dominated diffusion equation combined with Darcy's law is the widely used classical model. A lot of research and analysis have been done for the model, and also get a wide range of applications. Russell [53] has ana-lyzed the concentration equation by a combination of a Galerkin method and the method of characteristics and the press equation by a standard Galerkin procedure. Using the method of characteristics for solving the concentra-tion equation can make the equation symmetric, enhance stability, reduce the time truncation error, and the larger time step can be used. Pressure equation using the standard Galerkin method can only determine the pres-sure, but the speed can not be obtained directly. Ewing and Russell [28][25] have analyzed the coupled process by mixed finite elements and a modi-fied method of characteristics. This method can be obtained simultaneously by the pressure and velocity, without going through the differentiating the pressure to obtain an approximation of the velocity, thus reducing the speed error.
Another research focus of this article is the moving grid method. Us-ing the finite element method for solving the time-dependent problems (such as parabolic), the usually effective practice in in the region of space using the finite element method, and in the time direction using a finite difference scheme. A fixed set of finite element mesh have been proposed in the past algorithms which is limited to the region of space Grid. However, in many practical computing problems, we often need in different layers of time using different finite element spaces. For example, the flame pass Broadcast. and oil-water frontier. Therefore, many mathematicians and engineers focus on the dynamic finite element spare method, and many dynamic finite element methods have been proposed. Liang Guoping [39]proposed a moving grid finite element method for general parabolic problems. The main idea of the method is that a different number of finite element spaces is adopted at differ-ent time level, and the approximate solution at the current time is projected in the L2-norm onto the next time finite clement space and make it as an initial value. By developing the idea of coordinate transformation, he then presents a full-discretization moving finite element scheme[41], which can ap- ply to parabolic equations in any dimension with variable domain and under certain continuity assumptions about moving grid the optimal convergence rate is preserved. Yang Daoqi [61]proposed a mixed finite element method with moving grid for the parabolic problems, and extended to the porous medium miscible displacement [62]. Professor Yuan Yirang [65] discussed the moving grid method for nonlinear convection-diffusion problems.
This article is based on the finite element method of characteristics for the convection-diffusion equation. We mainly discussed the second or-der characteristics finite element method with moving grid, the second-order characteristics finite element method combined with the Galerkin finite ele-ment method in incompressible miscible displacement in porous media, and finally we gave an optimal-order error estimate for the mass-conservative characteristic finite clement scheme.
In Chapter1, we give the mathematical model which is discussed in this paper. According to the physical background of the miscible displace-ment in porous media, from the nature of mass conservation, we derive the convection-diffusion equation describing the concentration of fluids in porous media, and combining with Darcy equations describing fluid velocity and pressure relationship, we derive the coupled nonlinear partial differential equations. Then we introduce the sobolev spaces and its norms and give several lemmas frequently used in the later chapters.
In Chapter2, we disease the second order characteristics finite element method with moving grid. The approximation algorithm is of second order accuracy in time increment, and when the number of the moved grids M satisfies certain conditions, though it is not optimal accuracy in space incre-ment, but still have good convergence rate. When Mh is bounded, it has a convergence rate of hk in L2-norm.
In Chapter3, we develop the numerical analysis of incompressible mis-cible displacement in porous media, we use the second-order characteristics finite element method to treat the concentration equation and the Galerkin finite element method in the press equation. We list our assumptions on the problem, define the second order characteristic-Galerkin method and give the error estimates in L2-norm. Finally, we prove that the approximation algorithm is of second order accuracy in time increment, and though it is not optimal accuracy in space increment, but still have good convergence rate.
In Chapter3, an optimal-order error estimate is obtained for the model problem in one space dimension. The convergence rate is O(h2+Δt) in the l∞(L2) norm for the linear finite element. Some numerical examples are given to confirm the conclusion.
引文
[1]李荣华and冯果忱.偏微分方程数值解法.高等教育出版社,1996.
[2]孔祥言.高等渗流力学.中国科学技术大学出版社,合肥,2010.
[3]R. A. Adams and J. J. F. Fournier. Sobolev Spaces. Academic Press, Ams-terdam, second edition,2003.
[4]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.
[5]L. Badea, M. Discacciati, and A. Quarteroni. Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Mi-lan,2006.
[6]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.
[7]J. Bear. Dynamics of Fluids in Porous Media,Elsovier, New York,1972.
[8]J. Bear. Dynamics of Fluids in Porous Media. Elsevier, New York,1972.
[9]A. Bermudez. M. R. Nogueiras, and C. Vazqnez. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements, part i:time discretization. SIAM J. Numer. Anal.,44:1829-1853, 2006.
[10]A. Brmudez, M. R. Nogueiras, and C. Vazquez. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements, part ii:fully discretized scheme and quadrature formulas. SIAM J. Numer. Anal..44:1854-1876,2006.
[11]R. R. Bonnerot and P. Jamet. A second order finite element method for the one dimensional stefan problem. Int. J. Numer. Meth. Eng.,8:811-820,1974.
[12]S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York,1994.
[13]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.
[14]M. A. Celia, T. F. Russell, I. Herrera, and R. E. Ewing. An eulerian-lagrangian localized adjoint method for the advection-diffusion equation. Adv. Water Resources,13:187-206,1990.
[15]Z. Chen. Finite Element Methods and Their Applications. Scientific Compu-tation. Springer,2005.
[16]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.
[17]P. Ciarlet. The finite element method for elliptic problems. Amsterdam, North-Holland,1978.
[18]C. N. Dawson, T. F. Russell, and M. F. Wheeler. Some improved error estimates for the modified method of characteristics. SIAM J. Numer. Anal.. 120:1487-1552,1989.
[19]M. Discacciati. Domain decomposition methods for the coupling of surface and gronnduater flows. PhD thesis, Ecole Polytechnique Federale de lausanne, Switzerland,2004.
[20]J. Douglas, Jr., R. E. Ewing. and M. F. Wheeler. Approximation of the pres-sure by a mixed method in the simulation of miscible displacement. RAIRO Anal. Numer,17:17-33,1983.
[21]J. Douglas. Jr., R. E. Ewing, and M. F. Wheeler. A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numer,17:249-265,1983.
[22]J. Douglas, Jr., F. Furtado, and F. Pereira. On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Comput. Geosciences, 1:155-190,1997.
[23]J. Douglas, Jr., C.-S. Huang, and F. Pereira. The modified method of char-acteristics with adjusted advection. Numer. Math.,83:353-369,1999.
[24]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.
[25]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.
[26]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.
[27]R. E. Ewing, Y. Yuan, and G. Li. Timestepping 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.
[28]R.E. Ewing, T.F. Russell, and M.F. Wheeler. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite el-ements and a modified method of characteristics. Comput. Methods Appl. Mech. Engrg.,47:73-92,1984.
[29]R.E. Ewing and H. Wang. An optimal-order estimate for eulerian-lagrangian localized adjoint methods for variable-coeffirient advection-reaction problems. SIAM J. Numer. Anal.,33:318-348,199C.
[30]B.A. Finlayson. Numerical Methods for Problems with Moving Fronts. Ravenna Park Publishing, Seattle,1992.
[31]H. Fu and H. Rui. Characteristic-mixed methods on dynamic finite element spaces for two-phase miscible flow in porous media. Chinese Journal OF engineering mathematics,27:369-374,2010.
[32]A.0. 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.
[33]D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin,2001.
[34]D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin,2001.
[35]P. Jamet. Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal.,15:912-928, 1978.
[36]P. Jamet. Stability and convergence of a generalized crank-nicolson scheme on a variable mesh for the heat equation. SIAM J. Numer. Anal.,17:530-539, 1980.
[37]O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva. Liurar and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, RI,1968.
[38]R.J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge.2002.
[39]G. Liang. A fem with moving grid.J. Comput. Math.,4:377-384.1985
[40]G. Liang. A finite element method of semi-discretization with moving grid. J. Comput. Math.,4:86-96.1986.
[11]G. Liang and Z. Chen. A full-discretization moving fem with optimal conver-gence rate. J. Comput. Math..3:318-337.1990.
[42]Q. Lin and J. Lin. Finite element method:accuracy and Improvement. Science Press, Beijing,2006.
[43]J. L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York,1972.
[44]K. Miller and R. X. Miller. Moving finite elements. SIAM J. Numer. Anal, 18:1019-1057,1981.
[45]K.W. Morton. Numerical Solution of Convection-Diffusion Problems. Chap-man & Hall,1996.
[46]O. Pironneau, J. Liou, and T. Tezduyar. Characteristic-galerkin and galerkin/least-squares space-time formulations for the advection-diffusion equations with time-dependent domains. Comput. Methods Appl. Mech. En-grg.,100:117-141,1992.
[47]A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin,1994.
[48]R. Rannacher and R. Scott. Some optimal error estimates for piecewise linear finite element approximations. Math. Comp,28:437-445,1982.
[49]H. Rui. An alternative direction iterative method with second-order upwind scheme for convection-diffusion equations. Inter. J. Computer Math.,80:527-533,2003.
[50]H. Rui and M. Tabata. A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math.,92:161-177,2002.
[51]H. Rui and M. Tabata. A mass-conservative characteristic finite clement scheme for convection-diffusion problems. Journal of Scientific Computing, 43:416-432,2009.
[52]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.
[53]T. F. Russell. Time stepping along characteristics with incomplete iteration for a galerkin approximate of miscible displacement in porous media. SIAM J. Num.er. Anal.,22:970-1013,1985.
[54]E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton,1970.
[55]B. Straughan. Stability and Wave Motion in Porous Media. Springer, Berlin, 2008.
[56]V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin,1997.
[57]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.
[58]H. Wang, R.E. Ewing, and T.F. Russell. Eulerian-lagrangian localized meth-ods for convection-diffusion equations and their convergence analysis. SIAM J. Numer. Anal.,15:405-459,1995.
[59]K. Wang. A uniformly optimal-order error estimate of an ellam scheme for unsteady-state advection-diffusion equations. International Journal of Nu-merical Analysis and Modeling.5:286-302,2008.
[60]M. F. Wheeler. A priori l2 error estimates for galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal.,10:723-759, 1973.
[61]D. Yang. The mixed finite element methods with moving grids for parabolic problems. Mathematica Numcrica Sinica,10:266-271,1988.
[62]D. Yang. Mixed methods with dynamic finite-element spaces for miscible displacement in porous media. Journal of Computational and Applied Math-ematics,30:313-328.1990.
[63]D. Yang. A characteristic mixed method with dynamic finite element space for convection-dominated diffusion problems. Journal of Computational and Applied Mathematics,43:343-353,1992.
[64]K. Yosida. Functional Analysis. Springer-Verlag, New York. sixth edition, 1980.
[65]Y. Yuan. On characteristic finite element methods with moving mesh for nonlinear convection diffusion problems. Numerical Mathematics:A Journal of Chinese University,8:236-245,1983.
[2]孔祥言.高等渗流力学.中国科学技术大学出版社,合肥,2010.
[3]R. A. Adams and J. J. F. Fournier. Sobolev Spaces. Academic Press, Ams-terdam, second edition,2003.
[4]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.
[5]L. Badea, M. Discacciati, and A. Quarteroni. Mathematical analysis of the Navier-Stokes/Darcy coupling. Technical report, Politecnico di Milano, Mi-lan,2006.
[6]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.
[7]J. Bear. Dynamics of Fluids in Porous Media,Elsovier, New York,1972.
[8]J. Bear. Dynamics of Fluids in Porous Media. Elsevier, New York,1972.
[9]A. Bermudez. M. R. Nogueiras, and C. Vazqnez. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements, part i:time discretization. SIAM J. Numer. Anal.,44:1829-1853, 2006.
[10]A. Brmudez, M. R. Nogueiras, and C. Vazquez. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements, part ii:fully discretized scheme and quadrature formulas. SIAM J. Numer. Anal..44:1854-1876,2006.
[11]R. R. Bonnerot and P. Jamet. A second order finite element method for the one dimensional stefan problem. Int. J. Numer. Meth. Eng.,8:811-820,1974.
[12]S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York,1994.
[13]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.
[14]M. A. Celia, T. F. Russell, I. Herrera, and R. E. Ewing. An eulerian-lagrangian localized adjoint method for the advection-diffusion equation. Adv. Water Resources,13:187-206,1990.
[15]Z. Chen. Finite Element Methods and Their Applications. Scientific Compu-tation. Springer,2005.
[16]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.
[17]P. Ciarlet. The finite element method for elliptic problems. Amsterdam, North-Holland,1978.
[18]C. N. Dawson, T. F. Russell, and M. F. Wheeler. Some improved error estimates for the modified method of characteristics. SIAM J. Numer. Anal.. 120:1487-1552,1989.
[19]M. Discacciati. Domain decomposition methods for the coupling of surface and gronnduater flows. PhD thesis, Ecole Polytechnique Federale de lausanne, Switzerland,2004.
[20]J. Douglas, Jr., R. E. Ewing. and M. F. Wheeler. Approximation of the pres-sure by a mixed method in the simulation of miscible displacement. RAIRO Anal. Numer,17:17-33,1983.
[21]J. Douglas. Jr., R. E. Ewing, and M. F. Wheeler. A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numer,17:249-265,1983.
[22]J. Douglas, Jr., F. Furtado, and F. Pereira. On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Comput. Geosciences, 1:155-190,1997.
[23]J. Douglas, Jr., C.-S. Huang, and F. Pereira. The modified method of char-acteristics with adjusted advection. Numer. Math.,83:353-369,1999.
[24]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.
[25]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.
[26]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.
[27]R. E. Ewing, Y. Yuan, and G. Li. Timestepping 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.
[28]R.E. Ewing, T.F. Russell, and M.F. Wheeler. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite el-ements and a modified method of characteristics. Comput. Methods Appl. Mech. Engrg.,47:73-92,1984.
[29]R.E. Ewing and H. Wang. An optimal-order estimate for eulerian-lagrangian localized adjoint methods for variable-coeffirient advection-reaction problems. SIAM J. Numer. Anal.,33:318-348,199C.
[30]B.A. Finlayson. Numerical Methods for Problems with Moving Fronts. Ravenna Park Publishing, Seattle,1992.
[31]H. Fu and H. Rui. Characteristic-mixed methods on dynamic finite element spaces for two-phase miscible flow in porous media. Chinese Journal OF engineering mathematics,27:369-374,2010.
[32]A.0. 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.
[33]D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin,2001.
[34]D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin,2001.
[35]P. Jamet. Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal.,15:912-928, 1978.
[36]P. Jamet. Stability and convergence of a generalized crank-nicolson scheme on a variable mesh for the heat equation. SIAM J. Numer. Anal.,17:530-539, 1980.
[37]O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva. Liurar and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, RI,1968.
[38]R.J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge.2002.
[39]G. Liang. A fem with moving grid.J. Comput. Math.,4:377-384.1985
[40]G. Liang. A finite element method of semi-discretization with moving grid. J. Comput. Math.,4:86-96.1986.
[11]G. Liang and Z. Chen. A full-discretization moving fem with optimal conver-gence rate. J. Comput. Math..3:318-337.1990.
[42]Q. Lin and J. Lin. Finite element method:accuracy and Improvement. Science Press, Beijing,2006.
[43]J. L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York,1972.
[44]K. Miller and R. X. Miller. Moving finite elements. SIAM J. Numer. Anal, 18:1019-1057,1981.
[45]K.W. Morton. Numerical Solution of Convection-Diffusion Problems. Chap-man & Hall,1996.
[46]O. Pironneau, J. Liou, and T. Tezduyar. Characteristic-galerkin and galerkin/least-squares space-time formulations for the advection-diffusion equations with time-dependent domains. Comput. Methods Appl. Mech. En-grg.,100:117-141,1992.
[47]A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin,1994.
[48]R. Rannacher and R. Scott. Some optimal error estimates for piecewise linear finite element approximations. Math. Comp,28:437-445,1982.
[49]H. Rui. An alternative direction iterative method with second-order upwind scheme for convection-diffusion equations. Inter. J. Computer Math.,80:527-533,2003.
[50]H. Rui and M. Tabata. A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math.,92:161-177,2002.
[51]H. Rui and M. Tabata. A mass-conservative characteristic finite clement scheme for convection-diffusion problems. Journal of Scientific Computing, 43:416-432,2009.
[52]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.
[53]T. F. Russell. Time stepping along characteristics with incomplete iteration for a galerkin approximate of miscible displacement in porous media. SIAM J. Num.er. Anal.,22:970-1013,1985.
[54]E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton,1970.
[55]B. Straughan. Stability and Wave Motion in Porous Media. Springer, Berlin, 2008.
[56]V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin,1997.
[57]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.
[58]H. Wang, R.E. Ewing, and T.F. Russell. Eulerian-lagrangian localized meth-ods for convection-diffusion equations and their convergence analysis. SIAM J. Numer. Anal.,15:405-459,1995.
[59]K. Wang. A uniformly optimal-order error estimate of an ellam scheme for unsteady-state advection-diffusion equations. International Journal of Nu-merical Analysis and Modeling.5:286-302,2008.
[60]M. F. Wheeler. A priori l2 error estimates for galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal.,10:723-759, 1973.
[61]D. Yang. The mixed finite element methods with moving grids for parabolic problems. Mathematica Numcrica Sinica,10:266-271,1988.
[62]D. Yang. Mixed methods with dynamic finite-element spaces for miscible displacement in porous media. Journal of Computational and Applied Math-ematics,30:313-328.1990.
[63]D. Yang. A characteristic mixed method with dynamic finite element space for convection-dominated diffusion problems. Journal of Computational and Applied Mathematics,43:343-353,1992.
[64]K. Yosida. Functional Analysis. Springer-Verlag, New York. sixth edition, 1980.
[65]Y. Yuan. On characteristic finite element methods with moving mesh for nonlinear convection diffusion problems. Numerical Mathematics:A Journal of Chinese University,8:236-245,1983.