抛物问题的区域分解算法及特征线方法
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摘要
数学物理及工程问题,如油气藏的勘探与开发、大型结构工程、航天器的设计、天气预报、反应堆的计算等,无不归结为求解大型偏微分方程。计算区域往往是高维的、大范围的,其形态可能很不规则,给数值计算带来很大困难。区域分解算法是上世纪八十年代蓬勃发展起来的新方向。简单地说,区域分解算法是把计算区域Ω分解成若干子区域:(?),子区域Ω_i的形状尽可能规则,于是原问题的求解转化为在子区域上求解。由于区域分解算法能将大型问题分解为小型问题、复杂边值问题分解为简单边值问题、串行问题分解为并行问题,具有良好的并行性能。从上个世纪八十年代起,区域分解算法就已成为计算数学领域的研究热点。
区域分解算法大致可以分成两类:重叠型区域分解算法和非重叠型区域分解算法。子区域的选择主要依靠区域形状的可计算性以及问题的物理背景。尤其是后者,特别适用于在不同物理子区域上有不同控制方程的复合问题的计算。非重叠型区域分解算法实现起来比较直观易用,而重叠型区域分解算法的理论分析较为容易。
本文中,在区域分解算法方面,我们研究的重点是重叠型区域分解算法。重叠型区域分解算法是一种重要的求解偏微分方程的数值方法,其原始思想来源于经典的Schwarz交替法。近年来建立在Schwarz交替法基础上的区域分解算法在理论分析和实际应用中都取得了令人瞩目的发展,已成为一种有效的迭代方法。对于椭圆问题,许多基于重叠型区域分解的数值方法已经建立了系统的理论分析[6,14,16]。Lions[40,41]对热传导方程提出了一类建立在两个子区域基础上的Schwarz交替法,给出了收敛性结果,但没有给出误差估计。X.C.Cai [13,15,16,17]对于抛物问题构建了几类加性Schwarz算法和乘积性Schwarz算法,并证明了算法的收敛性,但作者没有详细讨论收敛率对离散参数的依赖性。芮洪兴教授和羊丹平教授[50,51]对于抛物问题论证了乘积性Schwarz算法的收敛性,并给出了误差估计与子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数之间的关系。
本文的另一个研究重点是特征线方法。近年来,随着科学技术的不断发展,关于对流-扩散方程求解问题的研究具有重要的理论和现实意义。对流-扩散方程可应用于油藏的运移聚集、油气资源的开采、地下水的污染、海水入侵等很多领域。从物理本质上考虑,它们都是流体在地下复杂的多孔介质间的运移,这种运移常常是对流占优或强对流占优的扩散运动。对于这些大规模长时间的实际应用问题,数值方法求解其数学模型常常遇到很大的困难。显著的困难在于对流占优的对流-扩散方程的解具有移动的陡峭前沿,求解一般抛物方程所用的标准有限元方法或有限差分方法会产生难以接受的数值震荡和数值弥散。为了克服数值震荡,工业中求解这类问题所使用的加权迎风方法又会产生较强的数值弥散。
J.Douglas,Jr.在1982年首次关于对流-扩散问题提出特征线方法[26]。将特征线方法引入到对流占优的对流-扩散问题中,这被认为是求解对流占优方程近似解的一种有效途径。此后,J.Douglas、Jr.、T.F.Ewing、M.F.Wheeler和袁益让教授等都对对流-扩散方程问题的研究和应用奠定了坚实的理论基础和实验基础[24,25,35,48,49,64,65]。从实验中可以看出,特征线方法相比一般的方法,能够增加时间步长,从而提高计算效率,同时避免数值震荡等优点,为大规模和大时间步长的数值计算提供了有效的途径。芮洪兴教授和M.Tabata在2002年建立了一种新的特征线方法[52]。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。
在导师芮洪兴教授的悉心指导下,本文作者在前人工作的基础上,对重叠型区域分解算法和特征线方法做了部分研究工作。首先,基于X.C.Cai提出的加性Schwarz算法,我们论证了抛物问题的加性Schwarz算法的收敛性以及误差估计对子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数的依赖性。理论分析和数值算例均表明,算法具有高度的并行性。其次,结合X.C.Cai提出的乘积性Schwarz算法和芮洪兴教授提出的二阶特征有限元方法(计算格式在时间步长上具有二阶精度),对对流-扩散方程构建了一种新的区域分解算法,并且理论上分析了算法的收敛性。最后,基于T.Arbogast和M.F.Wheeler建立的特征-混合元方法,我们给出一种特征-混合元格式来近似求解不稳定状态对流-扩散方程,此格式保持局部质量守恒性。本文作者在理论上证明了格式的收敛性,并且在一定的条件下论证了误差估计关于趋于零的扩散系数是一致的。
全文共分三章。
第一章,首先考虑由二阶抛物问题的Galerkin有限元方法所得到的线性代数方程组,我们在此基础上提出了加性Schwarz区域分解算法。此种算法的方程系数矩阵是对称正定的,可以应用共轭梯度法迭代求解。然后我们论证了抛物问题的加性Schwarz算法的收敛性及误差估计与子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数之间的关系。理论分析和数值算例均表明,算法具有高度的并行性。其次,我们引进一种修正的加性Schwarz算法,这类算法相比于一般加性Schwarz算法更适用于并行计算,并且同样在理论上证明了修正的加性Schwarz算法的收敛性及误差估计对子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数的依赖性。数值算例验证了算法的的有效性和优越性。本章的一些结果已经发表在《Applied Mathematics and Computation》上(见[46])。
第二章,主要研究对流一扩散方程的区域分解算法。我们知道,对流占优的对流-扩散方程的解具有移动的陡峭前沿,求解一般抛物方程所用的标准有限元方法或有限差分方法会产生难以接受的数值震荡和数值弥散。对此问题,采用芮洪兴教授所提出的二阶特征有限元格式(计算格式在时间步长上具有二阶精度)[52],并结合乘积性Schwarz算法,我们得到一种新的区域分解算法并且给出一个最优阶的误差估计。理论分析证明了算法的收敛性。我们还证明了误差估计与子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数之间的关系。
第三章,研究不稳定状态对流-扩散方程的特征-混合元方法。到目前为止,数值求解对流占优的对流-扩散方程仍然是计算数学专业的一个研究热点。近年来,这类数值方法的研究有很多,例如,显式特征线方法、流线扩散法[38]、修正的特征Galerkin有限元法(MMOC-Galerkin)[24,26,31]和Eulerian-Lagrangian局部共轭方法(ELLAM)[18,19,54,56,57]等等。每一种方法都有各自的优点和缺点。显式特征线方法需要一个Courant-Friedrich-lewy(CFL)时间步长条件。流线扩散法虽然能够在一定程度上消除数值弥散,但是增加了一个自定义的偏离于流线方向的量。对于修正的特征Galerkin有限元方法(MMOC-Galerkin),其优点是很大程度上消除了数值震荡和数值弥散,可用于大步长大规模数值计算(它是一种隐格式),并成功地应用于油藏问题的数值模拟。然而,这种方法不能保持物理问题所固有的质量守恒性。Euler-Lagrange局部共轭方法(ELLAM),作为改进的特征线方法不仅保持质量守恒性,而且能够方便地处理边界条件。但是在ELLAM方法中,计算所得到的积分是有一定困难的。在本章中,基于T.Arbogast和M.F.Wheeler首先提出的特征-混合元格式[2,3],我们给出不稳定状态对流-扩散方程的一个特征-混合元格式,此格式能够保持局部质量守恒性。最后给出一个一致的误差估计,也就是指得到的误差估计结果关于趋于零的扩散系数是一致的。
Mathematical physics and engineering problems can be turned into the problems of solving large partial differential equations. such as the exploration and development of oil and gas reservior, the large structure of engineering, the design of large spacecraft, aerodynamics, reactor etc.. The domains they are defined on are always large area with high dimension and irregular geometry, which cause many difficulties to computations when seeking their approximate solutions. Domain decomposition methods were new research direction which grew up in 1980s'. In short, domain decomposition methods divide the whole domainΩinto several sub-domains:(?), and the shapes of these sub-domainsΩ_i may as well as regular. And then the solutions of the original problems can be translated into solving the questions on the sub-domains respectively. Because they have many advantages, such as they can divide large-scale problems into several small ones, complex boundary problems into several simple boundary ones and successive problems into parallel ones, etc., which other methods don't have, they rapidly become the hot field of computational mathematics.
Domain decomposition methods may be partitioned into two families: overlapping domain decomposition methods and nonoverlapping domain decomposition methods. The selection of sub-domains may be based on considerations of available computing region shape and physical background of the problems. The latter is, in particular, applicable to complex systems which consist of possibly different governing equations in different physical sub-domains. Overlapping domain decomposition methods become harder to implement in such a setting, while the nonoverlapping domain decomposition methods may be more directly applicable. In addition, the theoretical analysis of the nonoverlapping domain decomposition methods are more difficult.
In this dissertation, our researches in domain decomposition methods are focussed on overlapping domain decomposition methods. The initial idea of overlapping domaindecomposition methods came from the classical Schwarz alternating algorithm. In recent years the theoretical researches and applications on domain decomposition methods based on Schwarz alternating algorithm have been developed adequately so that these methods become very powerful and efficient iterative methods. A systematic theory has been developed for elliptic finite element problems in the past few years [6, 14, 16]. Lions [40, 41] presented a kind of Schwarz alternating algorithm in two sub-domain case for heat equations and gave a convergence result but did not give any error estimate. X. C. Cai [13, 15, 16, 17] constructed a kind of additive Schwarz algorithms and multiplicative Schwarz algorithms and prove that the convergence rate is smaller than one for parabolic equations, where the author did not consider the dependence of the convergence and the discretization parameters. H. Rui and D. Yang [50, 51] considered how the convergence and the error estimate depend on the diameter of sub-domains,the spacial mesh-size, the time step increment and the number of iterations at each time level.
In this dissertation, we also do some researches on the method of characteristics for convection-diffusion equations. Convection-diffusion equations describe miscible displacement flow processes in petroleum reservoir simulation, subsurface contaminant transport and remediation, disposal of nuclear waste in underground repositories, and many other applications. Their solutions typically have moving steep fronts that need to be resolved accurately. Standard finite difference or Galerkin finite element methodstend to yield numerical solutions with severe nonphysical oscillations. The classical upwind finite difference method greatly reduces these oscillations but introduces excessivenumerical dispersion. Many extensive schemes have been carried out to overcome these difficulties and allow accurate numerical solutions with reasonable computational effort, such as the methods of characteristics [24, 25, 26, 35, 48, 49].
In 1982, J. Douglas, Jr. first developed the method of characteristics [26]. It is effective to approximate the solutions of convection-diffusion equations by using the method of characteristics. Since then, J. Douglas, Jr., R. E. Ewing, T. F. Russell, M. F. Wheeler, T. Arbogast and Y. Yuan have completed a series of fundamental researches in this field [24, 25, 35, 48, 49, 64, 65]. From the numerical experiments, we could find that characteristics methods could increase time step and avoid numerical oscillation, as opposed to general methods. In [52], Rui and Tabata first introduced a second order characteristic finite element method. The numerical scheme is of second order accuracy in time increment, symmetric and unconditionally stable.
Under the aborative guidance of my supervisor Professor Hongxing Rui, the authorhas finished this dissertation consisting of work on domain decomposition methods and characteristics methods. Firstly, based on additive Schwarz algorithms proposed by X. C. Cai, we present some additive Schwarz methods and analyze the convergenceof each procedure. And we study the dependence of the convergence rate on the spacial mesh size, the time increment, the sub-domains overlapping degree and the number of iterations at each time level. Both theoretical analysis and numerical experiments show that all these algorithms are high parallelism. Secondly, based on multiplicative Schwarz methods proposed by X. C. Cai and combined with a second order characteristic finite element scheme, a new kind of domain decomposition meth-ods are constructed for convection-diffusion equations. The analysis of stability and convergence of these algorithms is given. Finally, we apply a characteristics-mixed finite element method, which was proposed by T. Arbogast and M. F. Wheeler, to solve unsteady-state convection-diffusion equations. We prove the convergence of the scheme. The error estimate holds uniformly with respect to the vanishing diffusion coefficient.
The dissertation is divided into three chapters.
In Chapter 1, we firstly consider the solution of linear systems of algebraic equationsthat arise from parabolic finite element problems. We present additive Schwarz domain decomposition methods for parabolic problems and also consider the dependenceof convergence rates of these algorithms on parameters of time step and space-mesh.The resulting linear systems of equations are symmetric positive definite and solved by the conjugate gradient method. Both theoretical analysis and numerical experimentsshow that all these algorithms are high parallelism. Then, we introduce a kind of modified additive Schwarz methods also with the conjugate gradient method, which are more suitable for parallel computers than anterior algorithms. We analyze the convergence of modified additive Schwarz algorithms. Numerical experiments also confirm the efficiency and superiority of the algorithms. The part of the results in this chapter have been published in the Applied Mathematics and Computation (see [46]).
In Chapter 2, we mainly study the domain decomposition methods for convection- diffusion equations. We know that the solutions of convection-diffusion equations typicallyhave moving steep fronts. Standard finite difference or Galerkin finite element methods for parabolic problems tend to yield numerical solutions with severe non-physicaloscillations. Based on multiplicative Schwarz methods proposed by X. C. Cai and combined with a second order characteristic finite element scheme, a new kind of domain decomposition methods are constructed for convection-diffusion equations. The stability and convergence analysis of these algorithms is given. We analyze the dependence of convergence rates of these algorithms on parameters of time step and space-mesh.
In Chapter 3, we study a characteristics-mixed finite element method for unsteady-stateconvection-diffusion problems. The equation, which we consider in this chapter, is convection-dominated and the equation is nearly hyperbolic in nature. The concentrationoften develops sharp fronts that are nearly shocks. Because of the hyperbolic nature of convective transport, characteristic analysis is natural to aid in the solution of convection-diffusion equations. Effective discretization schemes recognize to some extent the hyperbolic nature of the equation. Many such schemes have been developed, such as the explicit method of characteristics, the streamline diffusion method [38], the modified method of characteristics-Galerkin finite element procedure (MMOC-Glerkin) [24, 26, 31], and the Eulerian-Lagrangian localized adjoint method (ELLAM) [18, 19, 54, 56, 57]. In this chapter, we develop a characteristics-mixed finite element method to solve unsteady-state convection-diffusion equations with all combinations of general inflow and outflow boundary conditions. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the convective (i.e., hyperbolic) part of the equation. Thus the scheme uses an approximate characteristics to handle advection in time. And we use a lowest-order Raviart-Thomas-Nedelec mixed finite element spatial approximation of the equation. Boundary conditions are incorporatedin a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. The error estimate holds uniformly with respect to the vanishing diffusioncoefficient.
区域分解算法大致可以分成两类:重叠型区域分解算法和非重叠型区域分解算法。子区域的选择主要依靠区域形状的可计算性以及问题的物理背景。尤其是后者,特别适用于在不同物理子区域上有不同控制方程的复合问题的计算。非重叠型区域分解算法实现起来比较直观易用,而重叠型区域分解算法的理论分析较为容易。
本文中,在区域分解算法方面,我们研究的重点是重叠型区域分解算法。重叠型区域分解算法是一种重要的求解偏微分方程的数值方法,其原始思想来源于经典的Schwarz交替法。近年来建立在Schwarz交替法基础上的区域分解算法在理论分析和实际应用中都取得了令人瞩目的发展,已成为一种有效的迭代方法。对于椭圆问题,许多基于重叠型区域分解的数值方法已经建立了系统的理论分析[6,14,16]。Lions[40,41]对热传导方程提出了一类建立在两个子区域基础上的Schwarz交替法,给出了收敛性结果,但没有给出误差估计。X.C.Cai [13,15,16,17]对于抛物问题构建了几类加性Schwarz算法和乘积性Schwarz算法,并证明了算法的收敛性,但作者没有详细讨论收敛率对离散参数的依赖性。芮洪兴教授和羊丹平教授[50,51]对于抛物问题论证了乘积性Schwarz算法的收敛性,并给出了误差估计与子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数之间的关系。
本文的另一个研究重点是特征线方法。近年来,随着科学技术的不断发展,关于对流-扩散方程求解问题的研究具有重要的理论和现实意义。对流-扩散方程可应用于油藏的运移聚集、油气资源的开采、地下水的污染、海水入侵等很多领域。从物理本质上考虑,它们都是流体在地下复杂的多孔介质间的运移,这种运移常常是对流占优或强对流占优的扩散运动。对于这些大规模长时间的实际应用问题,数值方法求解其数学模型常常遇到很大的困难。显著的困难在于对流占优的对流-扩散方程的解具有移动的陡峭前沿,求解一般抛物方程所用的标准有限元方法或有限差分方法会产生难以接受的数值震荡和数值弥散。为了克服数值震荡,工业中求解这类问题所使用的加权迎风方法又会产生较强的数值弥散。
J.Douglas,Jr.在1982年首次关于对流-扩散问题提出特征线方法[26]。将特征线方法引入到对流占优的对流-扩散问题中,这被认为是求解对流占优方程近似解的一种有效途径。此后,J.Douglas、Jr.、T.F.Ewing、M.F.Wheeler和袁益让教授等都对对流-扩散方程问题的研究和应用奠定了坚实的理论基础和实验基础[24,25,35,48,49,64,65]。从实验中可以看出,特征线方法相比一般的方法,能够增加时间步长,从而提高计算效率,同时避免数值震荡等优点,为大规模和大时间步长的数值计算提供了有效的途径。芮洪兴教授和M.Tabata在2002年建立了一种新的特征线方法[52]。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。
在导师芮洪兴教授的悉心指导下,本文作者在前人工作的基础上,对重叠型区域分解算法和特征线方法做了部分研究工作。首先,基于X.C.Cai提出的加性Schwarz算法,我们论证了抛物问题的加性Schwarz算法的收敛性以及误差估计对子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数的依赖性。理论分析和数值算例均表明,算法具有高度的并行性。其次,结合X.C.Cai提出的乘积性Schwarz算法和芮洪兴教授提出的二阶特征有限元方法(计算格式在时间步长上具有二阶精度),对对流-扩散方程构建了一种新的区域分解算法,并且理论上分析了算法的收敛性。最后,基于T.Arbogast和M.F.Wheeler建立的特征-混合元方法,我们给出一种特征-混合元格式来近似求解不稳定状态对流-扩散方程,此格式保持局部质量守恒性。本文作者在理论上证明了格式的收敛性,并且在一定的条件下论证了误差估计关于趋于零的扩散系数是一致的。
全文共分三章。
第一章,首先考虑由二阶抛物问题的Galerkin有限元方法所得到的线性代数方程组,我们在此基础上提出了加性Schwarz区域分解算法。此种算法的方程系数矩阵是对称正定的,可以应用共轭梯度法迭代求解。然后我们论证了抛物问题的加性Schwarz算法的收敛性及误差估计与子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数之间的关系。理论分析和数值算例均表明,算法具有高度的并行性。其次,我们引进一种修正的加性Schwarz算法,这类算法相比于一般加性Schwarz算法更适用于并行计算,并且同样在理论上证明了修正的加性Schwarz算法的收敛性及误差估计对子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数的依赖性。数值算例验证了算法的的有效性和优越性。本章的一些结果已经发表在《Applied Mathematics and Computation》上(见[46])。
第二章,主要研究对流一扩散方程的区域分解算法。我们知道,对流占优的对流-扩散方程的解具有移动的陡峭前沿,求解一般抛物方程所用的标准有限元方法或有限差分方法会产生难以接受的数值震荡和数值弥散。对此问题,采用芮洪兴教授所提出的二阶特征有限元格式(计算格式在时间步长上具有二阶精度)[52],并结合乘积性Schwarz算法,我们得到一种新的区域分解算法并且给出一个最优阶的误差估计。理论分析证明了算法的收敛性。我们还证明了误差估计与子区域长度、空间网格步长、时间步长和每一时间层上的迭代次数之间的关系。
第三章,研究不稳定状态对流-扩散方程的特征-混合元方法。到目前为止,数值求解对流占优的对流-扩散方程仍然是计算数学专业的一个研究热点。近年来,这类数值方法的研究有很多,例如,显式特征线方法、流线扩散法[38]、修正的特征Galerkin有限元法(MMOC-Galerkin)[24,26,31]和Eulerian-Lagrangian局部共轭方法(ELLAM)[18,19,54,56,57]等等。每一种方法都有各自的优点和缺点。显式特征线方法需要一个Courant-Friedrich-lewy(CFL)时间步长条件。流线扩散法虽然能够在一定程度上消除数值弥散,但是增加了一个自定义的偏离于流线方向的量。对于修正的特征Galerkin有限元方法(MMOC-Galerkin),其优点是很大程度上消除了数值震荡和数值弥散,可用于大步长大规模数值计算(它是一种隐格式),并成功地应用于油藏问题的数值模拟。然而,这种方法不能保持物理问题所固有的质量守恒性。Euler-Lagrange局部共轭方法(ELLAM),作为改进的特征线方法不仅保持质量守恒性,而且能够方便地处理边界条件。但是在ELLAM方法中,计算所得到的积分是有一定困难的。在本章中,基于T.Arbogast和M.F.Wheeler首先提出的特征-混合元格式[2,3],我们给出不稳定状态对流-扩散方程的一个特征-混合元格式,此格式能够保持局部质量守恒性。最后给出一个一致的误差估计,也就是指得到的误差估计结果关于趋于零的扩散系数是一致的。
Mathematical physics and engineering problems can be turned into the problems of solving large partial differential equations. such as the exploration and development of oil and gas reservior, the large structure of engineering, the design of large spacecraft, aerodynamics, reactor etc.. The domains they are defined on are always large area with high dimension and irregular geometry, which cause many difficulties to computations when seeking their approximate solutions. Domain decomposition methods were new research direction which grew up in 1980s'. In short, domain decomposition methods divide the whole domainΩinto several sub-domains:(?), and the shapes of these sub-domainsΩ_i may as well as regular. And then the solutions of the original problems can be translated into solving the questions on the sub-domains respectively. Because they have many advantages, such as they can divide large-scale problems into several small ones, complex boundary problems into several simple boundary ones and successive problems into parallel ones, etc., which other methods don't have, they rapidly become the hot field of computational mathematics.
Domain decomposition methods may be partitioned into two families: overlapping domain decomposition methods and nonoverlapping domain decomposition methods. The selection of sub-domains may be based on considerations of available computing region shape and physical background of the problems. The latter is, in particular, applicable to complex systems which consist of possibly different governing equations in different physical sub-domains. Overlapping domain decomposition methods become harder to implement in such a setting, while the nonoverlapping domain decomposition methods may be more directly applicable. In addition, the theoretical analysis of the nonoverlapping domain decomposition methods are more difficult.
In this dissertation, our researches in domain decomposition methods are focussed on overlapping domain decomposition methods. The initial idea of overlapping domaindecomposition methods came from the classical Schwarz alternating algorithm. In recent years the theoretical researches and applications on domain decomposition methods based on Schwarz alternating algorithm have been developed adequately so that these methods become very powerful and efficient iterative methods. A systematic theory has been developed for elliptic finite element problems in the past few years [6, 14, 16]. Lions [40, 41] presented a kind of Schwarz alternating algorithm in two sub-domain case for heat equations and gave a convergence result but did not give any error estimate. X. C. Cai [13, 15, 16, 17] constructed a kind of additive Schwarz algorithms and multiplicative Schwarz algorithms and prove that the convergence rate is smaller than one for parabolic equations, where the author did not consider the dependence of the convergence and the discretization parameters. H. Rui and D. Yang [50, 51] considered how the convergence and the error estimate depend on the diameter of sub-domains,the spacial mesh-size, the time step increment and the number of iterations at each time level.
In this dissertation, we also do some researches on the method of characteristics for convection-diffusion equations. Convection-diffusion equations describe miscible displacement flow processes in petroleum reservoir simulation, subsurface contaminant transport and remediation, disposal of nuclear waste in underground repositories, and many other applications. Their solutions typically have moving steep fronts that need to be resolved accurately. Standard finite difference or Galerkin finite element methodstend to yield numerical solutions with severe nonphysical oscillations. The classical upwind finite difference method greatly reduces these oscillations but introduces excessivenumerical dispersion. Many extensive schemes have been carried out to overcome these difficulties and allow accurate numerical solutions with reasonable computational effort, such as the methods of characteristics [24, 25, 26, 35, 48, 49].
In 1982, J. Douglas, Jr. first developed the method of characteristics [26]. It is effective to approximate the solutions of convection-diffusion equations by using the method of characteristics. Since then, J. Douglas, Jr., R. E. Ewing, T. F. Russell, M. F. Wheeler, T. Arbogast and Y. Yuan have completed a series of fundamental researches in this field [24, 25, 35, 48, 49, 64, 65]. From the numerical experiments, we could find that characteristics methods could increase time step and avoid numerical oscillation, as opposed to general methods. In [52], Rui and Tabata first introduced a second order characteristic finite element method. The numerical scheme is of second order accuracy in time increment, symmetric and unconditionally stable.
Under the aborative guidance of my supervisor Professor Hongxing Rui, the authorhas finished this dissertation consisting of work on domain decomposition methods and characteristics methods. Firstly, based on additive Schwarz algorithms proposed by X. C. Cai, we present some additive Schwarz methods and analyze the convergenceof each procedure. And we study the dependence of the convergence rate on the spacial mesh size, the time increment, the sub-domains overlapping degree and the number of iterations at each time level. Both theoretical analysis and numerical experiments show that all these algorithms are high parallelism. Secondly, based on multiplicative Schwarz methods proposed by X. C. Cai and combined with a second order characteristic finite element scheme, a new kind of domain decomposition meth-ods are constructed for convection-diffusion equations. The analysis of stability and convergence of these algorithms is given. Finally, we apply a characteristics-mixed finite element method, which was proposed by T. Arbogast and M. F. Wheeler, to solve unsteady-state convection-diffusion equations. We prove the convergence of the scheme. The error estimate holds uniformly with respect to the vanishing diffusion coefficient.
The dissertation is divided into three chapters.
In Chapter 1, we firstly consider the solution of linear systems of algebraic equationsthat arise from parabolic finite element problems. We present additive Schwarz domain decomposition methods for parabolic problems and also consider the dependenceof convergence rates of these algorithms on parameters of time step and space-mesh.The resulting linear systems of equations are symmetric positive definite and solved by the conjugate gradient method. Both theoretical analysis and numerical experimentsshow that all these algorithms are high parallelism. Then, we introduce a kind of modified additive Schwarz methods also with the conjugate gradient method, which are more suitable for parallel computers than anterior algorithms. We analyze the convergence of modified additive Schwarz algorithms. Numerical experiments also confirm the efficiency and superiority of the algorithms. The part of the results in this chapter have been published in the Applied Mathematics and Computation (see [46]).
In Chapter 2, we mainly study the domain decomposition methods for convection- diffusion equations. We know that the solutions of convection-diffusion equations typicallyhave moving steep fronts. Standard finite difference or Galerkin finite element methods for parabolic problems tend to yield numerical solutions with severe non-physicaloscillations. Based on multiplicative Schwarz methods proposed by X. C. Cai and combined with a second order characteristic finite element scheme, a new kind of domain decomposition methods are constructed for convection-diffusion equations. The stability and convergence analysis of these algorithms is given. We analyze the dependence of convergence rates of these algorithms on parameters of time step and space-mesh.
In Chapter 3, we study a characteristics-mixed finite element method for unsteady-stateconvection-diffusion problems. The equation, which we consider in this chapter, is convection-dominated and the equation is nearly hyperbolic in nature. The concentrationoften develops sharp fronts that are nearly shocks. Because of the hyperbolic nature of convective transport, characteristic analysis is natural to aid in the solution of convection-diffusion equations. Effective discretization schemes recognize to some extent the hyperbolic nature of the equation. Many such schemes have been developed, such as the explicit method of characteristics, the streamline diffusion method [38], the modified method of characteristics-Galerkin finite element procedure (MMOC-Glerkin) [24, 26, 31], and the Eulerian-Lagrangian localized adjoint method (ELLAM) [18, 19, 54, 56, 57]. In this chapter, we develop a characteristics-mixed finite element method to solve unsteady-state convection-diffusion equations with all combinations of general inflow and outflow boundary conditions. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the convective (i.e., hyperbolic) part of the equation. Thus the scheme uses an approximate characteristics to handle advection in time. And we use a lowest-order Raviart-Thomas-Nedelec mixed finite element spatial approximation of the equation. Boundary conditions are incorporatedin a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. The error estimate holds uniformly with respect to the vanishing diffusioncoefficient.
引文
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[2] T. Arbogast, A. Chilakapati, M. F. Wheeler, A characteristic-mixed method for contaminant transport and miscible displacement [C]. Computational Methods in Water Resources IX, Vol. 1: Numerical Methods in Water Resources (T. P. Russell, R. E. Ew-ing, C. A. Brebbia, W. G. Gray, G. F. Pindar), Computational Mechanics Publications, Southampton, U.K., 77-84(1992).
[3] T. Arbogast, M. F. Wheeler, A characteristics-mixed finite element method for advection-diffusion transport problems [J]. SIAM J. Numer. Anal., 32: 404-424(1995).
[4] L. Badea, I. R. Ionescu, S. Wolf, Schwarz method for earthquake source dynamics [J]. J. Comput. Phys., 227: 3824-3848(2008).
[5] S. Brunssen, B. Mohlmuth, An overlapping domain decomposition method for the simulation of elastoplastic incremental forming processes [J]. International Journal for Numerical Methods in Engineering, 77(9): 1224-1246(2009).
[6] J. H. Bramble, J. E. Pasciak, J. Wang, J. Xu, Convergence estimates for product iterative methods with application to domain decomposition [J]. Math. Comput., 57: 1-21(1991).
[7] J. H. Bramble, J. E. Pasciak, J. Xu, Parallel multilevel preconditioned [J]. Math. Comput., 55: 1-22(1990).
[8] F. Brezzi, J. Douglas, Jr., R. Duran, M. Fortin, Mixed finite elements for second order elliptic problems in three variables [J]. Number. Math., 51: 237-250(1987).
[9] F. Brezzi, J. Douglas, Jr., L. D. Marini, Two families of mixed finite elements for second order elliptic problems [J]. Numer. Math., 47: 217-235(1985).
[10] F. Brezzi, J. Douglas, Jr., M. Fortin, L. D. Marini, Efficient rectangular mixed finite elements in two and three space variables[J]. RAIRO Model. Math. Numer. Anal., 21: 581-603(1987).
[11] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods [M]. Springer-Verlag, Berlin, 1991.
[12] P. G. Ciarlet, The Finite Element Method for Elliptic Problems [M]. North-Holland, Amsterdam, 1987.
[13] X. C. Cai, Additive Schwarz algorithms for parabolic convection diffusion equations [J]. Numer. Math., 60: 41-61(1991).
[14] X. C. Cai, Additive Schwarz algorithms for nonselfadjoint elliptic equations [M]. Ph.D. thesis, Courant Institute, 1989.
[15] X. C. Cai, Multiplicative Schwarz algorithms for parabolic problem [J]. SIAM J. Sci. Comput., 15: 587-603(1994).
[16] X. C. Cai, Multiplicative Schwarz algorithms for nonsymmetric and indefinite problem [J]. SIAM J. Numer. Anal., 30: 936-952(1993).
[17] X. C. Cai, Some domain decomposition algorithms for nonselfadjoint elliptic and parabolic partial differential equations [M]. Ph. D. thesis, Courant Institute, 1989.
[18] M. A. Celia, T. F. Russell, I. Herrera, R. E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation [J]. Adv. Water Resources, 13: 187-206(1990)
[19] Z. Chen, J. Douglas, Jr., Prismatic mixed finite elements for second order elliptic problems [J]. Calcolo, 26: 135-148(1989).
[20] P. G. Ciarlet, The Finite Element Method for Elliptic Problems [M]. North-Holland, Amsterdam, 1987.
[21] L. C. Cowsar, T. F. Dupont, M. F. Wheeler, A priori estimates for mixed finite element methods for the wave equations [J]. Comput. Meth. Appl. Mech. Engrg., 82: 205-222(1990).
[22] L. C. Cowsar, T. F. Dupont, M. F. Wheeler, A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions [J]. SIAM J. Numer. Anal., 33: 492-504(1996).
[23] C. N. Dawson, Q. Du, T. F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equation [C]. Tech. Rep., Univ. of Chicago, 89-09(1989).
[24] C. N. Dawson, T. F. Russell, M. F. Wheeler, Some improved error estimates for the modified method of characteristics [J]. SIAM J. Numer. Anal., 26: 1487-1552(1989).
[25] L. Demkowicz, J. T. Oden, An adaptive characteristic Petro-Galerkin finite element scheme for convection-diffussion problems [J]. Comput. Meth. Appl. Mech. Engrg., 55: 63-87(1986).
[26] J. Douglas, Jr., T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures [J]. SIAM J. Numer. Anal., 19: 872-885(1982).
[27] M. Dryia, An additive Schwarz algorihm for two- and three- dimensional finite element elliptic problems [C]. Domain Decomposition Methods for Partial Differential Equations II(T. Chan, R. Glowinski, eds.), Philadelphia, 1989.
[28] M. Dryja and O. B. Widlund, An additive variant of Schwarz alternating method for the case of many subregions [C]. Tech. Rep. 339, Dept. Comput. Sci., Courant Institute, 1987.
[29] R. E. Duran, On the approximation of miscible displacement in porous media by a method of characteristic combined with a mixed method [J]. SIAM J. Numer. Anal., 25: 989-1001(1988).
[30] R. E. Ewing, R. D. Lazarov, J. E. Pasciak, P. S. Vassilevski, Finite element methods for parabolic problems with time steps variable in space [C]. Tech. Rep., 1989-05, Inst. for Sci. Comput., Univ. of Wyoming, 1989.
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