区域分解预处理器研究及其在地下水数值计算中的应用
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摘要
随着地下水数值模拟的深入研究,科研人员对于获取实时、精确、详细和可信任的地下水模拟信息的要求越来越高,对数值模拟软件能够针对具有精细网格剖分、长时间跨度特征问题进行模拟提出了迫切的需求。对研究区域的精细剖分往往导致数据占用内存多、求解效率低的问题。为了解决此类问题,一方面人们从数学模型出发,选择新的数值离散方法如(多尺度有限元、拉普拉斯变换有限层、FAC法等)通过减少剖分单元数来降低方程组维数进而降低内存的占用,在满足一定精度前提下提高求解效率；一方面针对数值离散后形成的大型、稀疏、病态的线性代数方程组,发展了多种高效的求解算法,其中预处理共轭梯度方法(PCG)已成为求解大型稀疏线性代数方程组极为有效的算法,而高效预处理器的构建是预处理共轭梯度方法的关键。
近年来,区域分解方法因其独特的优势备受关注。论文首先利用区域分解方法构建预处理器,给出区域分解预处理器(DDP, Domain Decomposition Preconditioner)实现的详细步骤,并与预处理共轭梯度方法结合成为区域分解预处理共轭梯度法(DDP-PCG)。将区域分解预处理共轭梯度法应用于一具有解析解的承压水流问题,验证了方法的可信性。对研究区进行不同规模剖分,分别采用CG、Jacobi-PCG、SSOR-PCG、DDP-PCG方法求解上述均质承压水问题,计算结果表明：在各种网格规模下,DDP-PCG的迭代次数均明显低于其他方法,并且其迭代次数几乎不随网格规模发生变化,说明了DDP-PCG方法具有较强的鲁棒性；然而DDP-PCG方法的CPU耗时却不是最低的。经分析上述现象正是由于子区域问题求解效率低引起的。
子区域上的快速算法是区域分解算法的基石。论文基于有限单元法,给出Dirichlet、Neumann边界11种组合模式下矩形区域均质承压水稳定流问题的傅里叶分析方法(FAM, Fourier Analysis method),对于傅里叶分析中的各种变换公式均采用快速傅里叶变换(FFT, Fast Fourier Transform)算法进行计算,并编制了相应的计算机程序,实现了均质承压水稳定流问题的快速求解。接着针对11种组合模式下的模型进行数值试验,将FAM方法的计算结果与解析解或者其他数值解对比,说明了FAM的可信性。采用存在解析解的单井定流量抽水承压水稳定流模型进行数值试验进一步验证了FAM的可信性,并且对该模型研究区进行多种不同模式剖分,分别采用FAM和迭代法(Jacobi、Gauss-Seidel、SOR、PCG)求解,计算结果表明：剖分精度越高,在求解精度相当的情况下FAM的求解效率越显著。此外针对均质承压水稳定流问题,采用FAM不需计算和存储原始系数矩阵,从而节省了大量内存空间。
对于DDP-PCG方法,当子区域问题采用FAM求解时,得到基于傅里叶分析的区域分解预处理共轭梯度法(A-DDP-PCG, DDP-PCG based on Fourier Analysis)。文中针对水文地质问题给出子区域划分模式、相应子区域对应的傅里叶分析方法,以及FA-DDP-PCG的求解步骤,并编制了相应的计算机程序。采用具有解析解的均质承压水模型验证了FA-DDP-PCG的可靠性。接着对该模型研究区进行不同规模网格剖分,均采用CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG求解,结果表明与CG, Jacobi-PCG, SSOR-PCG相比,在一定精度范围内,FA-DDP-PCG的求解效率更高；并且随着网格规模的增加,A-DDP-PCG的求解效率优势更加显著。对承压含水层介质参数连续变化和突变情况,进行大规模剖分,进一步证明FA-DDP-PCG的求解效率高于其它三种方法。对于介质参数连续变化的非均质模型,通过随机试验证明参数aijk(在水流模型中相当于介质的渗透系数或者导水系数)的取值对于算法求解效率影响并不明显,表明FA-DDP-PCG算法的健壮性。对于介质参数突变情形,研究了介质参数的差异性对算法求解效率的影响。结果表明,FA-DDP-PCG方法可以有效求解强突变介质地下水流问题。因此FA-DDP-PCG方法可以有效求解大规模地下水流问题。
With the in-depth study of the numerical simulation of groundwater, researchers make higher requirements for real-time, accurate, detailed and credible information of groundwater simulation, and have pressing needs for the numerical simulation software that can efficiently solve problems with the characteristics of fine meshes and long time span.The fine subdivision of the study area often leads to a higher memory and lower efficiency problem. In order to solve the mentioned problem, on the one hand, researchers select new discretization methods for the mathematical model, such as (multi-scale finite element method, Laplace transform finite layer method, and the FAC method, etc.) to reduce the equations dimension and memory storage with less meshes, and to improve the efficiency under certain precision. On the other hand, reaearchers pay much attention on efficient algorithms for solving the large, sparse, morbid linear algebraic equations. And so far the preconditioned conjugate gradient method (PCG) has already been an extremely efficient algorithm for solving large sparse linear algebraic equations. Furthermore, the construction of efficient preprocessor is the key for PCG.
In recent years, domain decomposition method is popular because of its unique advantages. In the paper, the preprocessor is built with domain decomposition methods and detailed implementation processes of the domain decomposition preprocessor (DDP) are given. Domain decomposition preconditioned conjugate gradient method (DDP-PCG) is made with the combination of DDP and PCG method. A confined groundwater problem with the analytical solution is used to verify the credibility of DDP-PCG. Under different discretizations of the study area,CG Jacobi-PCG, SSOR-PCG, DDP-PCG methods are used respectivly to solve the above-mentioned problem. Results show that the number of iterations of DDP-PCG are significantly lower than other methods in a variety of grid scale, and almost independent of the grid scale indicating the robustness of DDP-PCG. However, the CPU time-consuming of DDP-PCG is not the lowest which is caused by the low efficiency for solving the sub-region problems.
A fast algorithm for the sub-region problems is the cornerstone of the domain decomposition algorithm. Based on the finite element theory, fourier analysis method(FAM) is studied for the homogeneous confined steady flow problem in a rectangular area with11kinds of combination of the Dirichlet and Neumann boundary conditions. A variety of transformation formulas in the FAM are calculated using the FFT algorithm and the corresponding computer program are prepared to realize the fast solution for the homogeneous confined steady flow problems. Then, the results of FAM are compared to the analytical solution or other numerical solution for11numerical experiments to indicate the credibility of FAM. And the numerical experiment for the confined steady flow model with the single well of fiexed pumping flow rate which has the analytic solution further validates the credibility of FAM. Under different subdivision of the study area, FAM and iterative methods (Jacobi, Gauss-Seidel, SOR, PCG) are used to solve the above problem. And calculation results show that:under a certain precision, the finer the discretization of the study aera is, the more significant is the efficiency of FAM. In addition, for the homogeneous confined steady flow problem, FAM does not need the calculation and storage of the original coefficient matrix, which makes much savement of the memory space.
When the subregion problems of DDP-PCG are calculated with FAM, we get the domain decomposition preconditioned conjugate gradient method based on fourier analysis(FA-DDP-PCG). As for hydrogeological problems, the sub-region division modes, corresponding FAM for the subregions, as well as the solving steps of FA-DDP-PCG are given and corresponding computer program are prepared. FA-DDP-PCG is applied for homogeneous confined flow model with the analytical solution to verify the reliability of FA-DDP-PCG. Then CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG are used to solve the model under different subdivisions, which show that FA-DDP-PCG is of higher efficiency compared to other methods and the finer the subdivision is, the more notable is FA-DDP-PCG's efficiency under a certain precesion. Finally, confined aquifer with continuous coefficients and abrupt coefficients are caculated under finely subdivision, which further proves that FA-DDP-PCG's efficiency are the highest of all. For the case with continuous coefficients, randomized trials explaine that akij (which represents the permeability coefficient or transmissibility) has little effect on the algorithm's efficiency, which indicates FA-DDP-PCG is of strong robustness. For the case with abrupt coefficients, experiments with different parameters are caculated, which show that FA-DDP-PCG can effectively solve groundwater model with strongly abrupt coefficients. In all, FA-DDP-PCG is an effective method for solving large-scale groundwater flow problems.
近年来,区域分解方法因其独特的优势备受关注。论文首先利用区域分解方法构建预处理器,给出区域分解预处理器(DDP, Domain Decomposition Preconditioner)实现的详细步骤,并与预处理共轭梯度方法结合成为区域分解预处理共轭梯度法(DDP-PCG)。将区域分解预处理共轭梯度法应用于一具有解析解的承压水流问题,验证了方法的可信性。对研究区进行不同规模剖分,分别采用CG、Jacobi-PCG、SSOR-PCG、DDP-PCG方法求解上述均质承压水问题,计算结果表明：在各种网格规模下,DDP-PCG的迭代次数均明显低于其他方法,并且其迭代次数几乎不随网格规模发生变化,说明了DDP-PCG方法具有较强的鲁棒性；然而DDP-PCG方法的CPU耗时却不是最低的。经分析上述现象正是由于子区域问题求解效率低引起的。
子区域上的快速算法是区域分解算法的基石。论文基于有限单元法,给出Dirichlet、Neumann边界11种组合模式下矩形区域均质承压水稳定流问题的傅里叶分析方法(FAM, Fourier Analysis method),对于傅里叶分析中的各种变换公式均采用快速傅里叶变换(FFT, Fast Fourier Transform)算法进行计算,并编制了相应的计算机程序,实现了均质承压水稳定流问题的快速求解。接着针对11种组合模式下的模型进行数值试验,将FAM方法的计算结果与解析解或者其他数值解对比,说明了FAM的可信性。采用存在解析解的单井定流量抽水承压水稳定流模型进行数值试验进一步验证了FAM的可信性,并且对该模型研究区进行多种不同模式剖分,分别采用FAM和迭代法(Jacobi、Gauss-Seidel、SOR、PCG)求解,计算结果表明：剖分精度越高,在求解精度相当的情况下FAM的求解效率越显著。此外针对均质承压水稳定流问题,采用FAM不需计算和存储原始系数矩阵,从而节省了大量内存空间。
对于DDP-PCG方法,当子区域问题采用FAM求解时,得到基于傅里叶分析的区域分解预处理共轭梯度法(A-DDP-PCG, DDP-PCG based on Fourier Analysis)。文中针对水文地质问题给出子区域划分模式、相应子区域对应的傅里叶分析方法,以及FA-DDP-PCG的求解步骤,并编制了相应的计算机程序。采用具有解析解的均质承压水模型验证了FA-DDP-PCG的可靠性。接着对该模型研究区进行不同规模网格剖分,均采用CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG求解,结果表明与CG, Jacobi-PCG, SSOR-PCG相比,在一定精度范围内,FA-DDP-PCG的求解效率更高；并且随着网格规模的增加,A-DDP-PCG的求解效率优势更加显著。对承压含水层介质参数连续变化和突变情况,进行大规模剖分,进一步证明FA-DDP-PCG的求解效率高于其它三种方法。对于介质参数连续变化的非均质模型,通过随机试验证明参数aijk(在水流模型中相当于介质的渗透系数或者导水系数)的取值对于算法求解效率影响并不明显,表明FA-DDP-PCG算法的健壮性。对于介质参数突变情形,研究了介质参数的差异性对算法求解效率的影响。结果表明,FA-DDP-PCG方法可以有效求解强突变介质地下水流问题。因此FA-DDP-PCG方法可以有效求解大规模地下水流问题。
With the in-depth study of the numerical simulation of groundwater, researchers make higher requirements for real-time, accurate, detailed and credible information of groundwater simulation, and have pressing needs for the numerical simulation software that can efficiently solve problems with the characteristics of fine meshes and long time span.The fine subdivision of the study area often leads to a higher memory and lower efficiency problem. In order to solve the mentioned problem, on the one hand, researchers select new discretization methods for the mathematical model, such as (multi-scale finite element method, Laplace transform finite layer method, and the FAC method, etc.) to reduce the equations dimension and memory storage with less meshes, and to improve the efficiency under certain precision. On the other hand, reaearchers pay much attention on efficient algorithms for solving the large, sparse, morbid linear algebraic equations. And so far the preconditioned conjugate gradient method (PCG) has already been an extremely efficient algorithm for solving large sparse linear algebraic equations. Furthermore, the construction of efficient preprocessor is the key for PCG.
In recent years, domain decomposition method is popular because of its unique advantages. In the paper, the preprocessor is built with domain decomposition methods and detailed implementation processes of the domain decomposition preprocessor (DDP) are given. Domain decomposition preconditioned conjugate gradient method (DDP-PCG) is made with the combination of DDP and PCG method. A confined groundwater problem with the analytical solution is used to verify the credibility of DDP-PCG. Under different discretizations of the study area,CG Jacobi-PCG, SSOR-PCG, DDP-PCG methods are used respectivly to solve the above-mentioned problem. Results show that the number of iterations of DDP-PCG are significantly lower than other methods in a variety of grid scale, and almost independent of the grid scale indicating the robustness of DDP-PCG. However, the CPU time-consuming of DDP-PCG is not the lowest which is caused by the low efficiency for solving the sub-region problems.
A fast algorithm for the sub-region problems is the cornerstone of the domain decomposition algorithm. Based on the finite element theory, fourier analysis method(FAM) is studied for the homogeneous confined steady flow problem in a rectangular area with11kinds of combination of the Dirichlet and Neumann boundary conditions. A variety of transformation formulas in the FAM are calculated using the FFT algorithm and the corresponding computer program are prepared to realize the fast solution for the homogeneous confined steady flow problems. Then, the results of FAM are compared to the analytical solution or other numerical solution for11numerical experiments to indicate the credibility of FAM. And the numerical experiment for the confined steady flow model with the single well of fiexed pumping flow rate which has the analytic solution further validates the credibility of FAM. Under different subdivision of the study area, FAM and iterative methods (Jacobi, Gauss-Seidel, SOR, PCG) are used to solve the above problem. And calculation results show that:under a certain precision, the finer the discretization of the study aera is, the more significant is the efficiency of FAM. In addition, for the homogeneous confined steady flow problem, FAM does not need the calculation and storage of the original coefficient matrix, which makes much savement of the memory space.
When the subregion problems of DDP-PCG are calculated with FAM, we get the domain decomposition preconditioned conjugate gradient method based on fourier analysis(FA-DDP-PCG). As for hydrogeological problems, the sub-region division modes, corresponding FAM for the subregions, as well as the solving steps of FA-DDP-PCG are given and corresponding computer program are prepared. FA-DDP-PCG is applied for homogeneous confined flow model with the analytical solution to verify the reliability of FA-DDP-PCG. Then CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG are used to solve the model under different subdivisions, which show that FA-DDP-PCG is of higher efficiency compared to other methods and the finer the subdivision is, the more notable is FA-DDP-PCG's efficiency under a certain precesion. Finally, confined aquifer with continuous coefficients and abrupt coefficients are caculated under finely subdivision, which further proves that FA-DDP-PCG's efficiency are the highest of all. For the case with continuous coefficients, randomized trials explaine that akij (which represents the permeability coefficient or transmissibility) has little effect on the algorithm's efficiency, which indicates FA-DDP-PCG is of strong robustness. For the case with abrupt coefficients, experiments with different parameters are caculated, which show that FA-DDP-PCG can effectively solve groundwater model with strongly abrupt coefficients. In all, FA-DDP-PCG is an effective method for solving large-scale groundwater flow problems.
引文
[1]王浩,陈垂裕,秦大庸,等.地下水数值计算与应用研究进展综述[J].地学前缘,2010,17(6)：1-12.
[2]薛禹群,谢春红.地下水数值模拟[M].北京：科学出版社,2007.
[3]薛禹群,叶淑君,谢春红,等.多尺度有限元法在地下水模拟中的应用[J].水利学报,2004,(7)：7-13.
[4]林琳,杨金忠,方跃骏,等.多尺度有限元法在地下水三维数值模拟中的应用[J].中国农村水利水电,2005,(12)：10-12.
[5]江思珉,朱国荣,王浩然,等.FAC方法在地下水数值模拟中的应用[J].水利学报,2006,37(11)：1389-1392.
[6]刘运航,王旭东,诸宏博,等.承压含水层非稳定流拉普拉斯变换有限层分析[J].水利学报,2010,41(6)：748-753.
[7]Schaars F, Kamps P. MODGRID:Simultaneous solving of different groundwater flow models at various scales. In:eds. Proceedings of MODFLOW 2001 and Other Modeling Odysseys Conference, eds. Seo Poeter, Zheng and Poeter. Colorado School of Mines, Golden, Colorado, September,2001:11-14.
[8]Mehl S W, Hill M C. Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes[J]. Adv. Water. Resour. 2002,25(5):497-511.
[9]Mehl S W, Hill M C. Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes:a new method of intepolation and analysis of errors[J]. Adv. Water. Resour.2004,27 (9):899-912.
[10]Mehl S W, Hill M C. MODFLOW-2005, the US Geological Survey modular ground-water model——documentation of shared node local grid refinement(LGR) and the boundary flow and head(BFH) package[R]. U.S. G. S. Techniques and Mwthods Report 6-A12,2005.
[11]Mehl S W, Hill M C. MODFLOW-2000, the U.S.Geological Survey modular ground-water model-User guide to the link-AMG(LMG) package for solving matrix equations using an algebraic multigrid solver. U.S.G. S. Open-File Report 01-177.Denver, Colorado:USGS.2000.
[12]Hill C M. Solving groundwater flow problems by conjugate-gradient methods and the strongly implicit procedure[J]. Water Resour. Res.1990a,26 (9):1961-1969.
[13]Hill C M. Preconditioned conjugate-gradient 2(PCG2), a computer program for solving groundwater flow equations[R]. Water-Resources Investigations report 90-4048,U.S. G. S,1990b.
[14]Ashby S F, Falgout R D. A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations, best avaible copy from the ParFLOW project,1995.
[15]Saied F. Efficient Parallel Multigrid based Solvers for Large Scale Groundwater Flow Simulations[J]. Computers Math. Applic.1998,35(7):45-54.
[16]Detwiler R L, Mehl S and Rajaram H. et al. Comparison of an algebraic multigrid algorithm to two iterative solvers used for modeling groundwater flow and transport[J].Ground Water.2002, 40(3):267-272.
[17]程汤培.地下水流动数值模拟的高效并行计算研究.中国地质大学(北京)博士学位论文.2011.
[18]McDonald G M, Arlen W H. A modular three-dimensional finite-difference groundwater flow model. Reston,Virginia:the United States Geological Survey,1988.
[19]吕涛,石济民,林振宝.区域分解算法-偏微分方程数值解新技术[M].北京：科学出版社,1999.
[20]Meijerink J A, Van der Vorst H A. Incomplete cholesky and conjugate gradient with n extra diagonals[J].Math.Comput.1977,137(31):148-162.
[21]Meijerink J A, Van der Vorst H A. Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems[J]. J. Comput. Phy.1981, (44):134-155.
[22]Axelsson O. A survey of preconditioned interative methods for linear systems of algrbratic equations[J]. BIT.1985, (25):166-187.
[23]Axelsson O, Lindskog G. On the eigenvalue distribution of a class of preconditioning methods[J]. Numer. Math.1986, (48):479-498.
[24]Concus P, Colub G H, and Oleary D P. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. Preprint for Sparse Matrix Computations. Academic Press.1976:309-332.
[25]Concus P, Golub G H and Meurant G. Block preconditioning for the conjugate gradient method[J]. Siam J. Stat.Comput.1985,6(1):220-252.
[26]Saad Y. Practical use of polynomial preconditionings for the conjugate method[R]. Research Report. YALEU/DCS/RR-282.1984.
[27]Saad Y. ILUT:a dual threshold incomplete Lu factorization[J]. Numer. Linear. Algebra. Appl. 1994,1(4):387-402.
[28]Saad Y, Zhang J. BILUTM:A domain-based multi-level block ILUT preconditioner for general sparse matrices[J]. Siam J.Matri Anal.Appl.1998,21:279-299.
[29]Saad Y. Iterative methods for sparse linear systems[M], Soc. Ind. Math.2003.
[30]Chan T F, Vander V H A. Approximate and incomplete factorizations.Parallel Numerical Algorithms, ICASE/LaRC Interdisciplinary Series in Science and Engeneering.2001:167-202.
[31]Zhong Z B. Modified block SSOR preconditioners for symmetric positive definite linear systems[J]. Ann. Operat. Res.2001,103:263-282.
[32]张永杰,孙秦.预处理矩阵及其构造方法[J].长春理工大学学报.2006,29(4)：128-130.
[33]Bj(?)rstad P E, Wildlund O B. Iterative methods for the solution of elliptic problems on regions partitioned into substructures[J]. Siam J.Numer.Anal.1986,23(6):1097-1120.
[34]储德林,胡显承.非重叠型区域分解预处理共轭梯度法[J].计算数学.1993,(1)：58-68.
[35]Vereecken H O, Neuendorf G and Lindenmayr G et al. A Schwarz domain decomposition method for solution of transient unsaturated water flow on parallel computers[J]. Ecol. Mod.1996, 93 (1-3):275-289.
[36]Bramble J H, Pasciak J E and Schatz A H. An iterative method for Elliptic Problems on Regions Partitioned into Substructures[J]. Math Comp.1986,46:361-369.
[37]Bramble J H, Pasciak J E and Schatz A H. The Construction of Preconditioner for Elliptic Problrms by Substructuring I[J]. Math Comp.1986,47:103-134.
[38]Bramble J H Pasciak J E, Schatz A H. The Construction of Preconditioner for Elliptic Problrms by Substructuring II[J]. Math Comp.1987,49:1-16.
[39]Bramble J H Pasciak J E, Schatz A H. The Construction of Preconditioner for Elliptic Problrms by Substructuring Iv[J]. Math Comp.1989,53(187):1-24.
[40]Chan T F. Analysis of preconditioners for domain decomposition[J]. Siam J.Numer.Anal. 1987,24 (2):382-390.
[41]Dryja M. A capacitance matrix method for dirichlet problem on polygon region[J]. Numer. Math.1982,39:51-64.
[42]Dryja M. A finite element-capacitance matrix for elliptic problems on regions partitioned into subregions[J]. Numer. Math.1984,44:153-168.
[43]Dryia M. A Method of Domain Decomposition for 3-D Finite Element Problems, Domain Decomposition Methods for Partial Differential Equations[J]. SIAM. Philadelphis.1988,43-61.
[44]张胜,黄鸿慈.椭圆型方程的并行迭代区域分裂法-多子区域情形[J].计算数学.1991,(12)：1233-1241.
[45]张胜,黄鸿慈.椭圆型方程的并行迭代区域分裂法-两个子区域情形[J].计算数学.1992,(2)：240-248.
[46]张胜.关于二阶椭圆方程区域分裂法的最优预处理[J].计算数学.1993,(2)：235-241.
[47]胡显承,储德林.求解椭圆型问题的一种基于区域分解的预处理器[J].清华大学学报(自然科学版).1991,31(6)：12-19.
[48]胡显承,储德林.基于区域分解的椭圆型问题预处理器中区域边界算子的谱分析[J].清华大学学报(自然科学版).1992,32(3)：8-17.
[49]曹志浩.区域分解预条件的Fourier分析[J].高等学校计算数学学报.1992,(3)：223-258.
[50]储德林.代数多级网格预处理器(Ⅰ).清华大学学报(自然科学版).1992,32(6)：68-74.
[51]顾金生,胡显承.基于子结构法构造非协调元解椭圆型问题的预处理器(Ⅰ)[J].计算数学.1996,(2)：113-128.
[52]顾金生,胡显承.基于子结构法构造用非协调元椭圆形问题的预处理器Ⅰ[J].计算数学.1996,(2)：113-128.
[53]胡齐芽,梁国平.区域分解界面预条件子构造的一般框架[J].计算数学.1999,21(1)：117.128.
[54]Jenkins E W, Kelley C T and Miller C T., et al. An aggregration-based domain decomposition preconditioner for groundwater flow[M]. SIAM J. Sci.Comput.2011:430-441.
[55]薛禹群,朱学愚,吴吉春,等.地下水动力学原理[M].北京：地质出版社,1986.
[56]Harbaugh A W, Mcdonald G M. Programmer's documentation for Modflow, an update to the U.S. Geological Survey Modular Finite-Difference Ground water Flow Model[R].U.S.G.S., open-file report,1996:96-486.
[57]Harbaugh A W, Banta E R and Hill M C. et al. MODFLOW-2000, the U.S. Geological Survey modular ground-water model-User guide to modularization concepts and the Ground-Water Flow Process[R]. U.S.Geological Survey Open-File Report 00-92. Reston, Virginia:USGS.2000.
[58]武强,董东林,武刚,等.水资源评价的可视化专业软件(Visual Modflow)与应用潜力[J].水文地质工程地质.1999,(5)：21-23.
[59]贾金生,田冰,刘昌明Visual Modflow在地下水模拟中的应用——以河北省栾城为例[J].河北农业大学学报.2003,26(4)：71-78.
[60]祝晓彬.地下水模拟系统(GMS)软件[J].水文地下工程地质.2003,(5)：53-55.
[61]孔志浩,王开章,孔凡亮.基于FEFLOW的浅层地下水水环境演化模拟[J].中国农村水利水电.2009,(10)：5-7.
[62]卞锦宇,薛禹群,程诚,等.上海市浦西地区地下水三维数值模拟[J].中国岩溶.2002,21(3)：182-187.
[63]薛禹群,谢春红.水文地质学的数值法.北京：煤炭工业出版社,1980.
[64]Tyson H N, Weber E M. Groundwater management for Nations future-Computer simulation of groundwater basins[J]. J. Hydraul. Div. Amer. Soc. Civil Eng.1964,90(Hy4):198-216.
[65]张宏仁,李俊亭.有限差分法与有限单元法在渗流问题中的对比[J].水文地质工程地质.1979,(2)：50-56.
[66]张宏仁.解地下水渗流问题的有限差分法(一)[J].水文地质工程地质,1980：53-57.
[67]张宏仁.解地下水渗流问题的有限差分法(二)[J].水文地质工程地质,1980：54-57.
[68]张宏仁.解地下水渗流问题的有限差分法(三)[J].水文地质工程地质,1980：59-62.
[69]张宏仁.解地下水渗流问题的有限差分法(四)[J].水文地质工程地质,1980：56-61.
[70]张宏仁.不对称六边形网格及三向迭代解法[J].水文地质工程地质,1990：15-19.
[71]张宏仁.偏微分方程数值解法在地学应用中的对比分析[J].地质学报.1993,67(3)：266-274.
[72]任理.求解地下水流问题的混合拉普拉斯变换有限差分法[J].武汉水利电力大学学报.1993.26(5)：547-554.
[73]钱孝星,史小丽.用边界拟合坐标下的有限差分法评价城市的地下水资源[J].水利学报.1994,(4)：33-37.
[74]易连兴.解非均质各向异性地下水渗流模型的改进型有限差分法[J].地质论评.2007,53(6)：839-843.
[75]李树文,王永宣,康敏娟.基于灰色有限差分模拟的地下水位动态研究[J]地下水.2010, 32(4)：1-4.
[76]张宏仁.有限差分法的改进[J].水文地质工程地质,1994,(2)：27-30.
[77]李开泰,黄艾香,黄庆怀.有限元方法及其应用(Ⅰ)-方法构造和数学基础[M].西安：西安交通大学出版社,1984.
[78]Zienkiewica O C, Watson M and Cheung Y K. Stress analysis by finite wlement method-thermal effects[J]. Nucl. Eng.Des.1966,4(5):498-502.
[79]Javandel I, Withersp P A. Application of finite element method to transient flow in porous media[J]. Soc. Pet. Eng. J.1968,8(3):241-243.
[80]张宏仁.有限单元法的水头反常问题[J].水文地质工程地质.1992,(5)：22-24.
[81]Neumann S P, Narasimhan T N and Witherspoon P A. Application of mixed explicit-implicit finite element method to nonlinear diffusion-type problems, Finite Elements in Water resources. In: Proceedings of the first International Conference on Finite Elements in Water Resources.1976.
[82]林绍志,朱政嘉,郑义.用有限单元法评价区域地下水资源[J].长春地质学院学报.1983,(1)：67-79.
[83]吴毅强.用有限单元法评价大同盆地地下水资源[J].工程勘察.1987,(5)：36-40.
[84]唐仲华,陈崇希.求解地下水不稳定流问题的拉普拉斯变换有限元方法[J].湘潭矿业学院学报.1991,6(2)：123-129.
[85]王国利,陈天生.有限元法用于北方岩溶泉域地下水动态预测的研究[J].地质找矿论丛.1991,6(1)：95-101.
[86]李国敏,石钦周,马英杰.郑州北郊水源地地下水流三维有限元模拟[J].水文地质工程地质.1996,(5)：4-10.
[87]李存法,马自芬,刘秀婷.地下水非稳定渗流分析的显式有限单元法[J].地下水.2003,25(3)：150-153.
[88]李存法,刘秀婷,马自芬.地下水渗流分析问题的一种数值解法及其应用[J].水文地质工程地质.2004,(5)：72-73.
[89]汪家权,吴义锋,钱家忠,等.济南泉域岩溶地下水三维等参有限元数值模拟[J].煤田地质与勘探.2005,33(3)：39-41.
[90]杨晓芳,李玉倩,屈吉鸿.迁安盆地地下水资源有限元数值模拟与评价[J].地下水.2008,30(3)：12-15.
[91]王亮,朝伦巴根,刘小燕,等.有限元法在滦河内蒙蓝旗段中游地下水允许开采量评价中 的应用[J].中国农村水利水电.2010,(3)：36-42.
[92]马瑞杰,李欣.岩溶裂隙地下水流数学模型求解有限体积法及应用[J].吉林大学学报(地球科学版).2005,35(6)：762-765.
[93]尹哲.多孔介质渗流问题的对称有限体积方法.山东大学博士学位论文.2007.
[94]Hou T Y, Wu X H. A multiscale finite element method for elliptic problems in composite materials and porous media [J] J.Comput. Phys.1997,134:169-189.
[95]Hou T Y, Wu X H and Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients [J]. Math. Comput.1999,68(227):913-943.
[96]Ye S H, Xue Y Q and Xie C H. Application of the multiscale finite element method to flow in heterogeneous porous media[J]. Water Resour. Res.2004,40:1-9.
[97]叶淑君,吴吉春,薛禹群.多尺度有限元法求解非均质多孔介质中的三维地下水流问题[J].地球科学进展.2004,19(3)：437-442.
[98]Stanley S S, Myron B A and Puckett J. The finite layer method for groundwater flow models[J]. Water Resour. Res.1992,28(6):1715-1722.
[99]王文科,李俊亭,刘方.局部坐标下有限分析法及其在腊家滩水源地群孔抽水试验中的应用[J].西北地质.1995,16(4)：65-71.
[100]诸宏博,王旭东,宰金珉.承压含水层非稳定流有限层分析[J].工程勘查.2008,(8)：21-25.
[101]王旭东,殷宗泽,宰金珉,等.柱坐标系下地下水非稳定流模拟的有限层法[J].岩土工程学报.2009,31(1)：15-20.
[102]徐进,王旭东,刘运航.地下水向水平井三维流动的有限层分析[J].岩土力学.2011,32(3)：922-926.
[103]Kuiper L K. A comparison of the incomplete cholesky-conjugate gradient method with the strongly implicit method as applied to the solution of two-dimensional groundwater flow equations[J]. Water Resour. Res.1981,17(4):1082-1086.
[104]Kuiper L K. Computer program for solving ground-water flow equations by the preconditioned conjugate gradient method[R]. U.S.G.S. Water-Resources Investigations Report 87-4091.1987.
[105]Meyer P D, Valocchi A J and Ashby S F. et al. A numerical investigation of the conjugate gradient method as applied to three-dimensional groundwater flow problems in randomly heterogeneous porous media[J]. Water Resour. Res.1989,25(6):1440-1446.
[106]Mogensen K, Stenby E and Banerjee S et al. Comparison of Iterative Methods for Computing the Pressure Field in a Dynamic Network Model[J].Transp. Porous. Media.1999,37(3): 277-301.
[107]Osiensky J L, Williams R E. Potential Inaccuracies in MODFLOW Simulations Involving the SIP and SSOR Methods for Matrix solution[J]. Ground Water.1997,35(2):229-232.
[108]马驰.SIP和PCG2两种迭代法在地下水数值计算中的应用对比[J].西安科技学院学报.2002,22(1)：59-62.
[109]Mckeon T J, Chu W S. A multigrid model for steady flow in partially saturated porous media[J]. Water Resour. Res.1987,23(4):542-546.
[110]李铎.多重网格法及其与水文地质计算方法的比较[J].河北地质学院学报.1994,17(6)：556-560.
[111]Wilson J D, Naff R L.The US Geological Survey modular ground-water model-GMG linear equation solver package documentation.U.S. G S. Open-File Report[R].2004,1261:47.
[112]江思珉.多重网格算法在地下水数值模拟中的应用研究.南京大学博士学位论文.2007.
[113]Dong Y H, Li G M. A Parallel PCG Solver for MODFLOW[J]. Ground Water.2009, 47(6):845-850.
[114]徐腾.并行预处理共轭梯度法在内蒙能源基地地下水流场模型中的应用.中国地质大学(北京)硕士学位论文.2010.
[115]Willien F, Faille F and Schneider F. Domain decomposition methods applied to sedimentary basin modeling. Prco. Ninth Int. Conf. On domain decomposition Meths.1998, 736-744.
[116]Dan G C, Damien T D. Non-overlapping DDMs to solve flow in heterogeneous porous media. Proc. Fifteenth Int. Conf. On domain decomposition Meths.2003,:529-536.
[117]杨绍国,周熙襄.区域分解法地震正演模拟方法研究[J].地球物理学报.1996,39(5)：690-698.
[118]吕广忠,鞠斌山,栾志安.油藏水力压裂区域分解模拟算法[J].石油大学学报(自然科学版).1998,22(5)：61-67.
[119]舒继武,归丽忠,周维四,等.区域分解法解黑油数值模拟问题的并行计算[J].南 京大学学报(自然科学版).1999,35(1)：51-57.
[120]黄国荣,张瑾.复杂三维流动感数值模拟的非重叠区域分解法[J].水动力学研究与进展(A).2001,16(1)：44-50.
[121]周永霞,瞿建武,任安禄.粘性不可压缩三维流场分块算法研究[J].浙江大学学报(工学版).2002,36(1)：111-114.
[122]周驰,冯国泰,王松涛,等.涡轮叶栅气热耦合数值模拟[J].工程热物理学报.2003,24(2)：224-227.
[123]刘青昆,归丽忠,舒继武,等.区域分解法解黑油数值模拟问题改进的并行计算[J].南京大学学报(自然科学版).2003,39(2)：209-237.
[124]吴淑红,刘翔鹗,郭尚平.热才数值模拟中的区域分解算法[J].石油勘探与开发.2002,29(4)：110-112.
[125]冯天厚.区域分解方法在快速旋转行星流体动力学并行计算中的应用[J].天文学报.2007,48(3)：397-406.
[126]臧增亮,饶宜锐,潘晓滨,等.区域分解对气象模式并行计算速度的影响[J].计算机工程.2008,34(17)：4-6.
[127]张友良,冯夏庭,茹忠亮.基于区域分解算法的岩土大规模高性能并行有限元系统研究[J].岩石力学与工程学报.2004,23(21)：3636-3641.
[128]王浩然,朱国荣,江思珉,等.基于区域分解法的地下水有限元并行数值模拟[J].2005,41(3)：245-252.
[129]黄勇,周志芳.基于区域分解算法的地下水耦合模型及其应用[J].工程地质学报.2007,15(1)：103-107.
[130]王敏,朱国荣,王浩然,等.基于区域分解法的水文地质参数寻优研究[J].水文地质工程地质.2006,(1)：65-68.
[131]宋丽红,杨天行.边界元区域分解算法在地下水中的应用[J].世界地质.2002,21(1)：50-52.
[132]王浩然,朱国荣,赵金熙.基于区域分解法的地下水有限元与边界元耦合模型[J].地质论评.2003,49(1)：48-51.
[133]田玲玲.地下水渗流的直接边界元非重叠性区域分解算法.重庆大学硕士学位论文.2006.
[134]王敏,魏久传,郭建斌,等.区域分解算法在井田放水试验模拟中的应用[J].地下水. 2010,32(1)：1-3.
[135]Liggett J A, Liu P F. The boundary integral equation method for porous media flow[M]. Winchester, Mass.1983.
[136]王文科,李俊亭.地下水非稳定流的LTFAM算法[J].1993,15(4)：151-155.
[137]朱学愚,钱孝星.地下水水文学.北京：中国环境科学出版社,2005.
[138]李庆扬,王能超,易大义.数值分析[M].第4版.北京：清华大学出版社,施谱林格出版社,2001.
[139]陶文铨.数值传热学(第2版).西安：西安交通大学出版社,2001.
[140]孙讷正.地下水流的数学模型和数值方法.北京：地质出版社,1981.
[141]黄艾香,周天孝.有限元理论与方法.北京：科学出版社.2009.
[142]Hestenes M, Stiefel E. Methods of conjugate gradients for solving linear systems[J].J. Research NBS,1952,49:409-436.
[143]雷光耀.预处理技术与PCG算法[J].数学进展.1992,21(2)：129-138.
[144]Manteuffel T A. An incomplete factorization technique for positive definite linear systems[J]. Math. Comput.,1980,34(150):473-497.
[145]Chow E, Saad Y. Approximate inverse preconditioners for general sparse matrices. In Proc. of the Colorado Conference on interative Methods,1994.
[146]Benzi M, Meyer C D and Tuma M. A sparse approximate inverse preconditioner for the conjugate gradient method[J]. Siam J. Sci. Comput.1996,17(5):1135-1149.
[147]肖映雄,张平,尹久仁.求解一类网格结构模型的代数多重网格法[J].计算力学学报.2005,22(2)：176-182.
[148]Conrad V, Wallach Y. A faster SSOR algorythm[J]. Numer. Math.1977,27:371-372.
[149]Gustafsson I. A class of first order factorization methods. BIT.1978,18:147-156.
[150]Stathopoulos A. Preconditioned interative methods for the large、sparse、symmetric eigenvalue problem on multicomputers. A dissertation of Vanderbilt University.1995.
[151]Lee J, Zhang J and Lu C C. Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems[R]. Technical report No.342-02. Department of computer science, University of Kentucky, Lexington, KY, USA,2002.
[152]刘建新,崔翔,邵汉光.对ICCG算法的一种改进[J].华北电力学院学报.1994,2：1-5.
[153]张永杰,孙秦,李江海.大型稀疏线性方程组的改进ICCG方法[J].计算物理。2007,24(5)：581-584.
[154]雷光耀ICCG与MICCG的一种改进算法[J].应用数学学报.1992,15(2)：285-288.
[155]吴建平,王正华,李晓梅.带门槛不完全Cholesky分解存在的问题与改进[J].数值计算与计算机应用.2003,(3)：207-214.
[156]林绍忠.对称逐步超松弛预处理共轭梯度法的改进迭代格式[J].数值计算与计算机应用.1997,(4)：266-270.
[157]林绍忠.用预处理共轭梯度法求解有限元方程组及程序设计[J].河海大学学报.1998,26(3)：112-115.
[158]韩林,张子明,倪志强.应用改进算法的对称逐步超松弛预处理共轭梯度法进行大体积混凝土仿真计算[J].河海大学学报(自然科学版).2010,38(3)：278-283.
[159]薄亚明,徐剑飞.解泊松方程的快速预处理共轭梯度法[J].江南大学学报(自然科学版).2002,1(3)：218-224.
[160]王金铭,谢德馨,姚缨英.电磁场分析中大型实稀疏对称方程组的新的预处理解法[J].电工电能新技术.2002,21(1)：69-72.
[161]Zhang J.A grid-based multilevel incomplete LU factorization preconditioning technique for general sparse matrices[J]. Appl. Math. Comput.2001,124,95-115.
[162]张朝阳,孙炳楠,唐锦春.结构分析的并行预处理共轭梯度法[J].浙江大学学报(自然科学版).1997,31(3)：380-386.
[163]舒继武,赵金熙,张德富.解大型系数线性代数方程组的一种有效并行ICCG法[J].计算机工程与应用.1999,30-31.
[164]钱巍,张燕晖,迟媛.机群环境下基于PCG法的有限元并行算法[J].东北农业大学学报.2006,37(3)：390-392.
[165]黄志强.多项式-PCG并行算法研究.广东工业大学硕士学位论文.2004.
[166]Widlund O. Iterative methods for the solution of elliptic problems on regions partitioned into substructures and the biharmonoc Dirchlet problems. Proceedings of the sixth international conference on computing methods in applied and engineering held at Versailles, France, December 12-16,1983.
[167]Hilbert L B. Substructuring and domain decomposition techniques for analysis of discontinuous Media. A dissertation of the University of California at Berkeley.1995.
[168]Graham I G, Hagger M J. Unstructured additive schwars-conjugate gradient method for elliptic problems with highly discontinuous coefficients[J]. Siam. J. Sci. Comput.1999,20(6): 2041-2066.
[169]Zhu Y R. Domain decomposition preconditioners for elliptic equations with jump coefficients[J]. Numer. Linear. Algebra. Appl.2008,15:271-289.
[170]Hockney R W. A fast direct solution of possion's equation using Fourier analysis [J]. J. Associ. Comput. Mach.1965,12(1):95-113.
[171]Dorr F W. The direct solution of the discrete possion equation on a rectangle [J]. Siam Review,1970,12 (2):248-263.
[172]Swarztrauber P N. The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of passion equation on a rectangle [J]. Siam Review.1977,19(3): 490-501.
[173]孙德军,尹协远,庄礼贤.一类椭圆型方程的快速解法-ELLIP程序包简介[J].水动力学研究与进展,1993,8(1)：17-20.
[174]Cooley J W, Lewis P A W and Welch P D. The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine, and Laplace transforms [J]. J. Sound Vib.1970,12(3):315-337.
[175]李竞生,姚磊华.含水层参数识别方法[M],北京：地质出版社,2003.
[176]Chan Y K, Mullineux N and Reed J R. Analytic solutions for drawdowns in rectangular artesian aquifers [J]. J. Hydrol.1972,31(1-2):151-160.
[2]薛禹群,谢春红.地下水数值模拟[M].北京：科学出版社,2007.
[3]薛禹群,叶淑君,谢春红,等.多尺度有限元法在地下水模拟中的应用[J].水利学报,2004,(7)：7-13.
[4]林琳,杨金忠,方跃骏,等.多尺度有限元法在地下水三维数值模拟中的应用[J].中国农村水利水电,2005,(12)：10-12.
[5]江思珉,朱国荣,王浩然,等.FAC方法在地下水数值模拟中的应用[J].水利学报,2006,37(11)：1389-1392.
[6]刘运航,王旭东,诸宏博,等.承压含水层非稳定流拉普拉斯变换有限层分析[J].水利学报,2010,41(6)：748-753.
[7]Schaars F, Kamps P. MODGRID:Simultaneous solving of different groundwater flow models at various scales. In:eds. Proceedings of MODFLOW 2001 and Other Modeling Odysseys Conference, eds. Seo Poeter, Zheng and Poeter. Colorado School of Mines, Golden, Colorado, September,2001:11-14.
[8]Mehl S W, Hill M C. Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes[J]. Adv. Water. Resour. 2002,25(5):497-511.
[9]Mehl S W, Hill M C. Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes:a new method of intepolation and analysis of errors[J]. Adv. Water. Resour.2004,27 (9):899-912.
[10]Mehl S W, Hill M C. MODFLOW-2005, the US Geological Survey modular ground-water model——documentation of shared node local grid refinement(LGR) and the boundary flow and head(BFH) package[R]. U.S. G. S. Techniques and Mwthods Report 6-A12,2005.
[11]Mehl S W, Hill M C. MODFLOW-2000, the U.S.Geological Survey modular ground-water model-User guide to the link-AMG(LMG) package for solving matrix equations using an algebraic multigrid solver. U.S.G. S. Open-File Report 01-177.Denver, Colorado:USGS.2000.
[12]Hill C M. Solving groundwater flow problems by conjugate-gradient methods and the strongly implicit procedure[J]. Water Resour. Res.1990a,26 (9):1961-1969.
[13]Hill C M. Preconditioned conjugate-gradient 2(PCG2), a computer program for solving groundwater flow equations[R]. Water-Resources Investigations report 90-4048,U.S. G. S,1990b.
[14]Ashby S F, Falgout R D. A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations, best avaible copy from the ParFLOW project,1995.
[15]Saied F. Efficient Parallel Multigrid based Solvers for Large Scale Groundwater Flow Simulations[J]. Computers Math. Applic.1998,35(7):45-54.
[16]Detwiler R L, Mehl S and Rajaram H. et al. Comparison of an algebraic multigrid algorithm to two iterative solvers used for modeling groundwater flow and transport[J].Ground Water.2002, 40(3):267-272.
[17]程汤培.地下水流动数值模拟的高效并行计算研究.中国地质大学(北京)博士学位论文.2011.
[18]McDonald G M, Arlen W H. A modular three-dimensional finite-difference groundwater flow model. Reston,Virginia:the United States Geological Survey,1988.
[19]吕涛,石济民,林振宝.区域分解算法-偏微分方程数值解新技术[M].北京：科学出版社,1999.
[20]Meijerink J A, Van der Vorst H A. Incomplete cholesky and conjugate gradient with n extra diagonals[J].Math.Comput.1977,137(31):148-162.
[21]Meijerink J A, Van der Vorst H A. Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems[J]. J. Comput. Phy.1981, (44):134-155.
[22]Axelsson O. A survey of preconditioned interative methods for linear systems of algrbratic equations[J]. BIT.1985, (25):166-187.
[23]Axelsson O, Lindskog G. On the eigenvalue distribution of a class of preconditioning methods[J]. Numer. Math.1986, (48):479-498.
[24]Concus P, Colub G H, and Oleary D P. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. Preprint for Sparse Matrix Computations. Academic Press.1976:309-332.
[25]Concus P, Golub G H and Meurant G. Block preconditioning for the conjugate gradient method[J]. Siam J. Stat.Comput.1985,6(1):220-252.
[26]Saad Y. Practical use of polynomial preconditionings for the conjugate method[R]. Research Report. YALEU/DCS/RR-282.1984.
[27]Saad Y. ILUT:a dual threshold incomplete Lu factorization[J]. Numer. Linear. Algebra. Appl. 1994,1(4):387-402.
[28]Saad Y, Zhang J. BILUTM:A domain-based multi-level block ILUT preconditioner for general sparse matrices[J]. Siam J.Matri Anal.Appl.1998,21:279-299.
[29]Saad Y. Iterative methods for sparse linear systems[M], Soc. Ind. Math.2003.
[30]Chan T F, Vander V H A. Approximate and incomplete factorizations.Parallel Numerical Algorithms, ICASE/LaRC Interdisciplinary Series in Science and Engeneering.2001:167-202.
[31]Zhong Z B. Modified block SSOR preconditioners for symmetric positive definite linear systems[J]. Ann. Operat. Res.2001,103:263-282.
[32]张永杰,孙秦.预处理矩阵及其构造方法[J].长春理工大学学报.2006,29(4)：128-130.
[33]Bj(?)rstad P E, Wildlund O B. Iterative methods for the solution of elliptic problems on regions partitioned into substructures[J]. Siam J.Numer.Anal.1986,23(6):1097-1120.
[34]储德林,胡显承.非重叠型区域分解预处理共轭梯度法[J].计算数学.1993,(1)：58-68.
[35]Vereecken H O, Neuendorf G and Lindenmayr G et al. A Schwarz domain decomposition method for solution of transient unsaturated water flow on parallel computers[J]. Ecol. Mod.1996, 93 (1-3):275-289.
[36]Bramble J H, Pasciak J E and Schatz A H. An iterative method for Elliptic Problems on Regions Partitioned into Substructures[J]. Math Comp.1986,46:361-369.
[37]Bramble J H, Pasciak J E and Schatz A H. The Construction of Preconditioner for Elliptic Problrms by Substructuring I[J]. Math Comp.1986,47:103-134.
[38]Bramble J H Pasciak J E, Schatz A H. The Construction of Preconditioner for Elliptic Problrms by Substructuring II[J]. Math Comp.1987,49:1-16.
[39]Bramble J H Pasciak J E, Schatz A H. The Construction of Preconditioner for Elliptic Problrms by Substructuring Iv[J]. Math Comp.1989,53(187):1-24.
[40]Chan T F. Analysis of preconditioners for domain decomposition[J]. Siam J.Numer.Anal. 1987,24 (2):382-390.
[41]Dryja M. A capacitance matrix method for dirichlet problem on polygon region[J]. Numer. Math.1982,39:51-64.
[42]Dryja M. A finite element-capacitance matrix for elliptic problems on regions partitioned into subregions[J]. Numer. Math.1984,44:153-168.
[43]Dryia M. A Method of Domain Decomposition for 3-D Finite Element Problems, Domain Decomposition Methods for Partial Differential Equations[J]. SIAM. Philadelphis.1988,43-61.
[44]张胜,黄鸿慈.椭圆型方程的并行迭代区域分裂法-多子区域情形[J].计算数学.1991,(12)：1233-1241.
[45]张胜,黄鸿慈.椭圆型方程的并行迭代区域分裂法-两个子区域情形[J].计算数学.1992,(2)：240-248.
[46]张胜.关于二阶椭圆方程区域分裂法的最优预处理[J].计算数学.1993,(2)：235-241.
[47]胡显承,储德林.求解椭圆型问题的一种基于区域分解的预处理器[J].清华大学学报(自然科学版).1991,31(6)：12-19.
[48]胡显承,储德林.基于区域分解的椭圆型问题预处理器中区域边界算子的谱分析[J].清华大学学报(自然科学版).1992,32(3)：8-17.
[49]曹志浩.区域分解预条件的Fourier分析[J].高等学校计算数学学报.1992,(3)：223-258.
[50]储德林.代数多级网格预处理器(Ⅰ).清华大学学报(自然科学版).1992,32(6)：68-74.
[51]顾金生,胡显承.基于子结构法构造非协调元解椭圆型问题的预处理器(Ⅰ)[J].计算数学.1996,(2)：113-128.
[52]顾金生,胡显承.基于子结构法构造用非协调元椭圆形问题的预处理器Ⅰ[J].计算数学.1996,(2)：113-128.
[53]胡齐芽,梁国平.区域分解界面预条件子构造的一般框架[J].计算数学.1999,21(1)：117.128.
[54]Jenkins E W, Kelley C T and Miller C T., et al. An aggregration-based domain decomposition preconditioner for groundwater flow[M]. SIAM J. Sci.Comput.2011:430-441.
[55]薛禹群,朱学愚,吴吉春,等.地下水动力学原理[M].北京：地质出版社,1986.
[56]Harbaugh A W, Mcdonald G M. Programmer's documentation for Modflow, an update to the U.S. Geological Survey Modular Finite-Difference Ground water Flow Model[R].U.S.G.S., open-file report,1996:96-486.
[57]Harbaugh A W, Banta E R and Hill M C. et al. MODFLOW-2000, the U.S. Geological Survey modular ground-water model-User guide to modularization concepts and the Ground-Water Flow Process[R]. U.S.Geological Survey Open-File Report 00-92. Reston, Virginia:USGS.2000.
[58]武强,董东林,武刚,等.水资源评价的可视化专业软件(Visual Modflow)与应用潜力[J].水文地质工程地质.1999,(5)：21-23.
[59]贾金生,田冰,刘昌明Visual Modflow在地下水模拟中的应用——以河北省栾城为例[J].河北农业大学学报.2003,26(4)：71-78.
[60]祝晓彬.地下水模拟系统(GMS)软件[J].水文地下工程地质.2003,(5)：53-55.
[61]孔志浩,王开章,孔凡亮.基于FEFLOW的浅层地下水水环境演化模拟[J].中国农村水利水电.2009,(10)：5-7.
[62]卞锦宇,薛禹群,程诚,等.上海市浦西地区地下水三维数值模拟[J].中国岩溶.2002,21(3)：182-187.
[63]薛禹群,谢春红.水文地质学的数值法.北京：煤炭工业出版社,1980.
[64]Tyson H N, Weber E M. Groundwater management for Nations future-Computer simulation of groundwater basins[J]. J. Hydraul. Div. Amer. Soc. Civil Eng.1964,90(Hy4):198-216.
[65]张宏仁,李俊亭.有限差分法与有限单元法在渗流问题中的对比[J].水文地质工程地质.1979,(2)：50-56.
[66]张宏仁.解地下水渗流问题的有限差分法(一)[J].水文地质工程地质,1980：53-57.
[67]张宏仁.解地下水渗流问题的有限差分法(二)[J].水文地质工程地质,1980：54-57.
[68]张宏仁.解地下水渗流问题的有限差分法(三)[J].水文地质工程地质,1980：59-62.
[69]张宏仁.解地下水渗流问题的有限差分法(四)[J].水文地质工程地质,1980：56-61.
[70]张宏仁.不对称六边形网格及三向迭代解法[J].水文地质工程地质,1990：15-19.
[71]张宏仁.偏微分方程数值解法在地学应用中的对比分析[J].地质学报.1993,67(3)：266-274.
[72]任理.求解地下水流问题的混合拉普拉斯变换有限差分法[J].武汉水利电力大学学报.1993.26(5)：547-554.
[73]钱孝星,史小丽.用边界拟合坐标下的有限差分法评价城市的地下水资源[J].水利学报.1994,(4)：33-37.
[74]易连兴.解非均质各向异性地下水渗流模型的改进型有限差分法[J].地质论评.2007,53(6)：839-843.
[75]李树文,王永宣,康敏娟.基于灰色有限差分模拟的地下水位动态研究[J]地下水.2010, 32(4)：1-4.
[76]张宏仁.有限差分法的改进[J].水文地质工程地质,1994,(2)：27-30.
[77]李开泰,黄艾香,黄庆怀.有限元方法及其应用(Ⅰ)-方法构造和数学基础[M].西安：西安交通大学出版社,1984.
[78]Zienkiewica O C, Watson M and Cheung Y K. Stress analysis by finite wlement method-thermal effects[J]. Nucl. Eng.Des.1966,4(5):498-502.
[79]Javandel I, Withersp P A. Application of finite element method to transient flow in porous media[J]. Soc. Pet. Eng. J.1968,8(3):241-243.
[80]张宏仁.有限单元法的水头反常问题[J].水文地质工程地质.1992,(5)：22-24.
[81]Neumann S P, Narasimhan T N and Witherspoon P A. Application of mixed explicit-implicit finite element method to nonlinear diffusion-type problems, Finite Elements in Water resources. In: Proceedings of the first International Conference on Finite Elements in Water Resources.1976.
[82]林绍志,朱政嘉,郑义.用有限单元法评价区域地下水资源[J].长春地质学院学报.1983,(1)：67-79.
[83]吴毅强.用有限单元法评价大同盆地地下水资源[J].工程勘察.1987,(5)：36-40.
[84]唐仲华,陈崇希.求解地下水不稳定流问题的拉普拉斯变换有限元方法[J].湘潭矿业学院学报.1991,6(2)：123-129.
[85]王国利,陈天生.有限元法用于北方岩溶泉域地下水动态预测的研究[J].地质找矿论丛.1991,6(1)：95-101.
[86]李国敏,石钦周,马英杰.郑州北郊水源地地下水流三维有限元模拟[J].水文地质工程地质.1996,(5)：4-10.
[87]李存法,马自芬,刘秀婷.地下水非稳定渗流分析的显式有限单元法[J].地下水.2003,25(3)：150-153.
[88]李存法,刘秀婷,马自芬.地下水渗流分析问题的一种数值解法及其应用[J].水文地质工程地质.2004,(5)：72-73.
[89]汪家权,吴义锋,钱家忠,等.济南泉域岩溶地下水三维等参有限元数值模拟[J].煤田地质与勘探.2005,33(3)：39-41.
[90]杨晓芳,李玉倩,屈吉鸿.迁安盆地地下水资源有限元数值模拟与评价[J].地下水.2008,30(3)：12-15.
[91]王亮,朝伦巴根,刘小燕,等.有限元法在滦河内蒙蓝旗段中游地下水允许开采量评价中 的应用[J].中国农村水利水电.2010,(3)：36-42.
[92]马瑞杰,李欣.岩溶裂隙地下水流数学模型求解有限体积法及应用[J].吉林大学学报(地球科学版).2005,35(6)：762-765.
[93]尹哲.多孔介质渗流问题的对称有限体积方法.山东大学博士学位论文.2007.
[94]Hou T Y, Wu X H. A multiscale finite element method for elliptic problems in composite materials and porous media [J] J.Comput. Phys.1997,134:169-189.
[95]Hou T Y, Wu X H and Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients [J]. Math. Comput.1999,68(227):913-943.
[96]Ye S H, Xue Y Q and Xie C H. Application of the multiscale finite element method to flow in heterogeneous porous media[J]. Water Resour. Res.2004,40:1-9.
[97]叶淑君,吴吉春,薛禹群.多尺度有限元法求解非均质多孔介质中的三维地下水流问题[J].地球科学进展.2004,19(3)：437-442.
[98]Stanley S S, Myron B A and Puckett J. The finite layer method for groundwater flow models[J]. Water Resour. Res.1992,28(6):1715-1722.
[99]王文科,李俊亭,刘方.局部坐标下有限分析法及其在腊家滩水源地群孔抽水试验中的应用[J].西北地质.1995,16(4)：65-71.
[100]诸宏博,王旭东,宰金珉.承压含水层非稳定流有限层分析[J].工程勘查.2008,(8)：21-25.
[101]王旭东,殷宗泽,宰金珉,等.柱坐标系下地下水非稳定流模拟的有限层法[J].岩土工程学报.2009,31(1)：15-20.
[102]徐进,王旭东,刘运航.地下水向水平井三维流动的有限层分析[J].岩土力学.2011,32(3)：922-926.
[103]Kuiper L K. A comparison of the incomplete cholesky-conjugate gradient method with the strongly implicit method as applied to the solution of two-dimensional groundwater flow equations[J]. Water Resour. Res.1981,17(4):1082-1086.
[104]Kuiper L K. Computer program for solving ground-water flow equations by the preconditioned conjugate gradient method[R]. U.S.G.S. Water-Resources Investigations Report 87-4091.1987.
[105]Meyer P D, Valocchi A J and Ashby S F. et al. A numerical investigation of the conjugate gradient method as applied to three-dimensional groundwater flow problems in randomly heterogeneous porous media[J]. Water Resour. Res.1989,25(6):1440-1446.
[106]Mogensen K, Stenby E and Banerjee S et al. Comparison of Iterative Methods for Computing the Pressure Field in a Dynamic Network Model[J].Transp. Porous. Media.1999,37(3): 277-301.
[107]Osiensky J L, Williams R E. Potential Inaccuracies in MODFLOW Simulations Involving the SIP and SSOR Methods for Matrix solution[J]. Ground Water.1997,35(2):229-232.
[108]马驰.SIP和PCG2两种迭代法在地下水数值计算中的应用对比[J].西安科技学院学报.2002,22(1)：59-62.
[109]Mckeon T J, Chu W S. A multigrid model for steady flow in partially saturated porous media[J]. Water Resour. Res.1987,23(4):542-546.
[110]李铎.多重网格法及其与水文地质计算方法的比较[J].河北地质学院学报.1994,17(6)：556-560.
[111]Wilson J D, Naff R L.The US Geological Survey modular ground-water model-GMG linear equation solver package documentation.U.S. G S. Open-File Report[R].2004,1261:47.
[112]江思珉.多重网格算法在地下水数值模拟中的应用研究.南京大学博士学位论文.2007.
[113]Dong Y H, Li G M. A Parallel PCG Solver for MODFLOW[J]. Ground Water.2009, 47(6):845-850.
[114]徐腾.并行预处理共轭梯度法在内蒙能源基地地下水流场模型中的应用.中国地质大学(北京)硕士学位论文.2010.
[115]Willien F, Faille F and Schneider F. Domain decomposition methods applied to sedimentary basin modeling. Prco. Ninth Int. Conf. On domain decomposition Meths.1998, 736-744.
[116]Dan G C, Damien T D. Non-overlapping DDMs to solve flow in heterogeneous porous media. Proc. Fifteenth Int. Conf. On domain decomposition Meths.2003,:529-536.
[117]杨绍国,周熙襄.区域分解法地震正演模拟方法研究[J].地球物理学报.1996,39(5)：690-698.
[118]吕广忠,鞠斌山,栾志安.油藏水力压裂区域分解模拟算法[J].石油大学学报(自然科学版).1998,22(5)：61-67.
[119]舒继武,归丽忠,周维四,等.区域分解法解黑油数值模拟问题的并行计算[J].南 京大学学报(自然科学版).1999,35(1)：51-57.
[120]黄国荣,张瑾.复杂三维流动感数值模拟的非重叠区域分解法[J].水动力学研究与进展(A).2001,16(1)：44-50.
[121]周永霞,瞿建武,任安禄.粘性不可压缩三维流场分块算法研究[J].浙江大学学报(工学版).2002,36(1)：111-114.
[122]周驰,冯国泰,王松涛,等.涡轮叶栅气热耦合数值模拟[J].工程热物理学报.2003,24(2)：224-227.
[123]刘青昆,归丽忠,舒继武,等.区域分解法解黑油数值模拟问题改进的并行计算[J].南京大学学报(自然科学版).2003,39(2)：209-237.
[124]吴淑红,刘翔鹗,郭尚平.热才数值模拟中的区域分解算法[J].石油勘探与开发.2002,29(4)：110-112.
[125]冯天厚.区域分解方法在快速旋转行星流体动力学并行计算中的应用[J].天文学报.2007,48(3)：397-406.
[126]臧增亮,饶宜锐,潘晓滨,等.区域分解对气象模式并行计算速度的影响[J].计算机工程.2008,34(17)：4-6.
[127]张友良,冯夏庭,茹忠亮.基于区域分解算法的岩土大规模高性能并行有限元系统研究[J].岩石力学与工程学报.2004,23(21)：3636-3641.
[128]王浩然,朱国荣,江思珉,等.基于区域分解法的地下水有限元并行数值模拟[J].2005,41(3)：245-252.
[129]黄勇,周志芳.基于区域分解算法的地下水耦合模型及其应用[J].工程地质学报.2007,15(1)：103-107.
[130]王敏,朱国荣,王浩然,等.基于区域分解法的水文地质参数寻优研究[J].水文地质工程地质.2006,(1)：65-68.
[131]宋丽红,杨天行.边界元区域分解算法在地下水中的应用[J].世界地质.2002,21(1)：50-52.
[132]王浩然,朱国荣,赵金熙.基于区域分解法的地下水有限元与边界元耦合模型[J].地质论评.2003,49(1)：48-51.
[133]田玲玲.地下水渗流的直接边界元非重叠性区域分解算法.重庆大学硕士学位论文.2006.
[134]王敏,魏久传,郭建斌,等.区域分解算法在井田放水试验模拟中的应用[J].地下水. 2010,32(1)：1-3.
[135]Liggett J A, Liu P F. The boundary integral equation method for porous media flow[M]. Winchester, Mass.1983.
[136]王文科,李俊亭.地下水非稳定流的LTFAM算法[J].1993,15(4)：151-155.
[137]朱学愚,钱孝星.地下水水文学.北京：中国环境科学出版社,2005.
[138]李庆扬,王能超,易大义.数值分析[M].第4版.北京：清华大学出版社,施谱林格出版社,2001.
[139]陶文铨.数值传热学(第2版).西安：西安交通大学出版社,2001.
[140]孙讷正.地下水流的数学模型和数值方法.北京：地质出版社,1981.
[141]黄艾香,周天孝.有限元理论与方法.北京：科学出版社.2009.
[142]Hestenes M, Stiefel E. Methods of conjugate gradients for solving linear systems[J].J. Research NBS,1952,49:409-436.
[143]雷光耀.预处理技术与PCG算法[J].数学进展.1992,21(2)：129-138.
[144]Manteuffel T A. An incomplete factorization technique for positive definite linear systems[J]. Math. Comput.,1980,34(150):473-497.
[145]Chow E, Saad Y. Approximate inverse preconditioners for general sparse matrices. In Proc. of the Colorado Conference on interative Methods,1994.
[146]Benzi M, Meyer C D and Tuma M. A sparse approximate inverse preconditioner for the conjugate gradient method[J]. Siam J. Sci. Comput.1996,17(5):1135-1149.
[147]肖映雄,张平,尹久仁.求解一类网格结构模型的代数多重网格法[J].计算力学学报.2005,22(2)：176-182.
[148]Conrad V, Wallach Y. A faster SSOR algorythm[J]. Numer. Math.1977,27:371-372.
[149]Gustafsson I. A class of first order factorization methods. BIT.1978,18:147-156.
[150]Stathopoulos A. Preconditioned interative methods for the large、sparse、symmetric eigenvalue problem on multicomputers. A dissertation of Vanderbilt University.1995.
[151]Lee J, Zhang J and Lu C C. Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems[R]. Technical report No.342-02. Department of computer science, University of Kentucky, Lexington, KY, USA,2002.
[152]刘建新,崔翔,邵汉光.对ICCG算法的一种改进[J].华北电力学院学报.1994,2：1-5.
[153]张永杰,孙秦,李江海.大型稀疏线性方程组的改进ICCG方法[J].计算物理。2007,24(5)：581-584.
[154]雷光耀ICCG与MICCG的一种改进算法[J].应用数学学报.1992,15(2)：285-288.
[155]吴建平,王正华,李晓梅.带门槛不完全Cholesky分解存在的问题与改进[J].数值计算与计算机应用.2003,(3)：207-214.
[156]林绍忠.对称逐步超松弛预处理共轭梯度法的改进迭代格式[J].数值计算与计算机应用.1997,(4)：266-270.
[157]林绍忠.用预处理共轭梯度法求解有限元方程组及程序设计[J].河海大学学报.1998,26(3)：112-115.
[158]韩林,张子明,倪志强.应用改进算法的对称逐步超松弛预处理共轭梯度法进行大体积混凝土仿真计算[J].河海大学学报(自然科学版).2010,38(3)：278-283.
[159]薄亚明,徐剑飞.解泊松方程的快速预处理共轭梯度法[J].江南大学学报(自然科学版).2002,1(3)：218-224.
[160]王金铭,谢德馨,姚缨英.电磁场分析中大型实稀疏对称方程组的新的预处理解法[J].电工电能新技术.2002,21(1)：69-72.
[161]Zhang J.A grid-based multilevel incomplete LU factorization preconditioning technique for general sparse matrices[J]. Appl. Math. Comput.2001,124,95-115.
[162]张朝阳,孙炳楠,唐锦春.结构分析的并行预处理共轭梯度法[J].浙江大学学报(自然科学版).1997,31(3)：380-386.
[163]舒继武,赵金熙,张德富.解大型系数线性代数方程组的一种有效并行ICCG法[J].计算机工程与应用.1999,30-31.
[164]钱巍,张燕晖,迟媛.机群环境下基于PCG法的有限元并行算法[J].东北农业大学学报.2006,37(3)：390-392.
[165]黄志强.多项式-PCG并行算法研究.广东工业大学硕士学位论文.2004.
[166]Widlund O. Iterative methods for the solution of elliptic problems on regions partitioned into substructures and the biharmonoc Dirchlet problems. Proceedings of the sixth international conference on computing methods in applied and engineering held at Versailles, France, December 12-16,1983.
[167]Hilbert L B. Substructuring and domain decomposition techniques for analysis of discontinuous Media. A dissertation of the University of California at Berkeley.1995.
[168]Graham I G, Hagger M J. Unstructured additive schwars-conjugate gradient method for elliptic problems with highly discontinuous coefficients[J]. Siam. J. Sci. Comput.1999,20(6): 2041-2066.
[169]Zhu Y R. Domain decomposition preconditioners for elliptic equations with jump coefficients[J]. Numer. Linear. Algebra. Appl.2008,15:271-289.
[170]Hockney R W. A fast direct solution of possion's equation using Fourier analysis [J]. J. Associ. Comput. Mach.1965,12(1):95-113.
[171]Dorr F W. The direct solution of the discrete possion equation on a rectangle [J]. Siam Review,1970,12 (2):248-263.
[172]Swarztrauber P N. The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of passion equation on a rectangle [J]. Siam Review.1977,19(3): 490-501.
[173]孙德军,尹协远,庄礼贤.一类椭圆型方程的快速解法-ELLIP程序包简介[J].水动力学研究与进展,1993,8(1)：17-20.
[174]Cooley J W, Lewis P A W and Welch P D. The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine, and Laplace transforms [J]. J. Sound Vib.1970,12(3):315-337.
[175]李竞生,姚磊华.含水层参数识别方法[M],北京：地质出版社,2003.
[176]Chan Y K, Mullineux N and Reed J R. Analytic solutions for drawdowns in rectangular artesian aquifers [J]. J. Hydrol.1972,31(1-2):151-160.