地下水问题及基于POD方法的降维模式研究
|
摘要
土壤水连接地表水与地下水,并通过影响陆面与大气之间的水分和能量交换影响气候,在气候与陆地水循环系统中起着十分重要的作用.地下水的动态变化直接影响地表土壤湿度导致地表感热潜热发生改变并对气候产生重要影响,反过来地表的入渗或蒸发通过影响土壤中水分含量引起地下水位的上升或下降,因而土壤水和地下水有着重要的联系.完全的三维土壤水地下水耦合模型,因其在大区域数值求解过程出现的庞大计算量,使得在气候与陆地水循环模式中合理表示这种相互作用较难实现.依据大尺度陆面过程土壤水与地下水主要特征,发展同时考虑土壤水垂直流与地下水水平流的拟三维地下水模型,简化了计算量,使得在气候与陆地水循环模式中合理表示土壤水和地下水相互作用成为可能.
在由二维或高维偏微分方程控制的物理模型中,数值离散求解是最有效的方法,但对方程离散得到的代数方程组的自由度太多,这需要很大计算量.特征正交分解(Proper Orthogonal Decomposition,简记为POD)方法是一种能提供适当逼近而自由度又较少的高效降维方法.降低计算维数、减少计算量和节省计算时间是POD方法最明显的优点.将POD与常用数值解法相结合,构造出一种既保持数值解适当精度又有较少自由度的降维数值格式,在未来的数值计算中有广阔的应用前景.
本博士论文的研究目标是建立用于气候与陆地水循环系统模拟的拟三维地下水模型,并发展二维土壤水模型基于特征正交分解的降维数值解法,以提高数值模型计算效率.围绕上述研究目标,本论文在考虑地下水与土壤水相互作用拟三维模型的建立、评估及应用,以及POD优化数值算法等方面开展研究,论文主要成果总结如下：
(1)建立了河流输水条件下考虑土壤水和地下水相互作用拟二维地下水模型.河流输水条件下的地下水埋深及土壤含水量的估计在河流剖面上归结为考虑土壤水与地下水相互作用的二维运动边界问题；将其土壤水流分解为以垂向流为主的非饱和土壤水运动和以侧向流为主的地下水运动,发展了考虑河流输水条件下土壤水垂直运动和地下水水平流动的土壤水与地下水相互作用拟二维数值模型；通过对模型的河水水位、水平导水率和地表通量参数的敏感性试验及分析,并结合SCE-UA参数优化方法对塔里木河下游生态输水下的英苏断面地下水位进行模拟验证,结果表明该模型能比较合理地模拟地下水位变化情况.
(2)建立了考虑土壤水和地下水相互作用的拟三维地下水模型.区域上土壤水与地下水相互作用问题可归结为三维饱和与非饱和流运动边界问题；将其土壤水流分解成以垂直流为主的一维土壤水运动和以水平流为主的二维地下水运动,发展了土壤水和地下水相互作用的拟三维地下水数值模型,模型由水平方向二维地下水数值模型、基于水平离散网格的垂直一维土壤水流模型和水平网格内两者的联系方程构成；同时采用自适应网格加密方法提出了一种新的土壤垂直分层结构来计算各层节点的土壤含水量；通过理想试验对模型的河水水位、地表水力传导率和地表通量进行敏感性分析以及结合SCE-UA参数优化方法,对塔里木河下游英苏断面上的地下水位进行了模拟验证,表明模型具有在不同条件下对地下水埋深的模拟能力；同时对塔里木河流域地下水位进行数值模拟试验,检验了地下水侧向流对大尺度起伏地形与河流水条件的响应,结果表明模型具有模拟大尺度地下水侧向流动的能力.
(3)发展了二维土壤水模型基于POD的降维数值解法.针对二维非饱和土壤水流方程问题,发展分别基于隐式差分格式和有限元格式的特征正交分解(POD)降维格式,它们具有计算量很少且足够高精度的特点,并给出其简化的POD隐式差分格式和POD有限元格式解的误差分析；数值例子表明在简化的POD隐式差分格式和POD有限元格式与通常的差分格式和有限元格式解之间的误差足够小的情况下,POD隐式差分格式和有限元格式比通常的差分格式和有限元格式大大地节省计算量,从而验证POD方法的有效性.
Soil water connecting surface water and groundwater affects climate by influenc-ing the water and energy exchange between land surface and atmosphere, and it plays a important role in climate and water circulation system. Dynamic variations in the water table over a region directly influence soil moisture at the surface, which leads to effects on latent and sensible heat fluxes and the growth and development of natural vegetation over the region. Water table fluctuations are influenced by surface fluxes such as infil-tration or evapotranspiration through the unsaturated zone, thus there is an important connection between soil water and ground water.
It is difficult to represent reasonably the interaction between soil water and ground water in climate and land circulation model with the fully three-dimensional soil wa-ter and groundwater model. There are many computational quantities when the three-dimensional interaction model is applied to large-scale regions. According to the main character of soil water flow and ground water flow in land surface processes, we have developed a quasi three-dimensional groundwater model including the equations of one-dimensional vertical soil water flow and two-dimensional horizonal groundwater flow. The model reduces computation and make it possible that representing reasonably the interaction of soil water and groundwater in climate and land water circulation model.
The numerical computation is the most effective method in solving the two-dimensi onal or high-dimensional partial difference equation. however, the discrete equations ob-tained by numerical method tend to be more unknown variables, the computation will be time-consuming. Proper orthogonal decomposition (POD) is an effective method for ap-proximating a large amount of data. Reducing computational dimensions、diminishing computational quantity and saving computation time are the most obvious advantage of POD. Combining the POD and usually numerical computation method and constructing a kind of numeric format with high accuracy and lower free degrees will be the broad application prospects.
Research objective of this paper is building a quasi three-dimensional ground-water model suitable for the simulation of the climate and land water circulation system, and developing a kind of dimension-reduction numerical solution based on POD for the two-dimensional soil water model, so as to improve numerical compu- tation efficiency. The main content of this paper includes as follows:
(1) Built a quasi two-dimensional numerical model based on soil water-groundwater and stream-aquifer interactions. The estimation of water table depth and soil water content on the soil profile under stream-aquifer interaction was reduced to be the two-dimensional moving boundary problem with soil water-groundwater interaction. The soil water flow was approximately divided into an unsaturated vertical soil water flow and a horizontal groundwater flow for the simulation of the interaction between soil water and groundwater. We developed a quasi two-dimensional numeric model based on vertical soil water equation and horizontal water table equation under stream-aquifer interaction. The sensitivities of the main parameters of the developed model were dis-cussed and its validation at the Yingsu Section of the lower reaches of the Tarim River with stream water conveyance showed the model can simulate water table dynamic vari-ation reasonably.
(2) Built a quasi three-dimensional variably saturated groundwater numeri-cal model based on soil water-groundwater interaction. The interaction of soil water and groundwater was reduced to be a three-dimensional saturated-unsaturated soil water flows problem. We developed a quasi three-dimensional, variably saturated groundwater flow model by approximately dividing the three-dimensional soil water and groundwater flow into an unsaturated vertical soil water flow and a horizontal groundwater flow for simulation of the interactions between soil water and groundwater. For each unsaturated soil column, a new soil layer structure was presented and an adaptive grid refinement method was applied to simulate the soil moisture content at the cell nodes. Synthetic ex-periments using the quasi three-dimensional numerical model were conducted to test the sensitivities of the main parameters of the model, including the river elevation, ground surface horizontal hydraulic conductivity, and surface flux. The results showed the ro-bustness of the developed model under different conditions. The estimation scheme of water table depth based the developed quasi three-dimensional model and SCE-UA pa-rameter calibration method was validated by a case of stream water conveyance in the lower reaches of the Tarim River. The simulation results of the water table elevations were in good agreement with the observations in the four monitoring wells. The param-eter test and model application for the prediction of water table elevations at the Yingsu section in the lower reaches of the Tarim River showed that it can be more reasonable to use the developed model with an adaptive refinement grid to substitute the original ten-layer unsaturated soil water structure. A numerical experiment by the model for Tarim River Basin was conducted to test the model on simulation of the effects of lateral groundwater flow for large-scale high-relief topography with stream water conveyance. The results showed that the model had the potential to be used in large-scale groundwa-ter latent flow.
(3)Developed dimension-reduce numerical method based on POD for the two-dimensional unsaturated soil water flow equation. For the two-dimensional unsatu-rated soil water equation, an optimizing reduced implicit difference scheme (IDS) and reduced finite element scheme (FES) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) was presented, respectively. The error esti-mates of the corresponding reduced POD numerical scheme were given and proved. The numerical examples showed the error between the reduced POD solution and usu-ally numerical solution was small enough, and computation time was reduced. Thus, both the feasibility and efficiency of our reduced IDS were validated.
在由二维或高维偏微分方程控制的物理模型中,数值离散求解是最有效的方法,但对方程离散得到的代数方程组的自由度太多,这需要很大计算量.特征正交分解(Proper Orthogonal Decomposition,简记为POD)方法是一种能提供适当逼近而自由度又较少的高效降维方法.降低计算维数、减少计算量和节省计算时间是POD方法最明显的优点.将POD与常用数值解法相结合,构造出一种既保持数值解适当精度又有较少自由度的降维数值格式,在未来的数值计算中有广阔的应用前景.
本博士论文的研究目标是建立用于气候与陆地水循环系统模拟的拟三维地下水模型,并发展二维土壤水模型基于特征正交分解的降维数值解法,以提高数值模型计算效率.围绕上述研究目标,本论文在考虑地下水与土壤水相互作用拟三维模型的建立、评估及应用,以及POD优化数值算法等方面开展研究,论文主要成果总结如下：
(1)建立了河流输水条件下考虑土壤水和地下水相互作用拟二维地下水模型.河流输水条件下的地下水埋深及土壤含水量的估计在河流剖面上归结为考虑土壤水与地下水相互作用的二维运动边界问题；将其土壤水流分解为以垂向流为主的非饱和土壤水运动和以侧向流为主的地下水运动,发展了考虑河流输水条件下土壤水垂直运动和地下水水平流动的土壤水与地下水相互作用拟二维数值模型；通过对模型的河水水位、水平导水率和地表通量参数的敏感性试验及分析,并结合SCE-UA参数优化方法对塔里木河下游生态输水下的英苏断面地下水位进行模拟验证,结果表明该模型能比较合理地模拟地下水位变化情况.
(2)建立了考虑土壤水和地下水相互作用的拟三维地下水模型.区域上土壤水与地下水相互作用问题可归结为三维饱和与非饱和流运动边界问题；将其土壤水流分解成以垂直流为主的一维土壤水运动和以水平流为主的二维地下水运动,发展了土壤水和地下水相互作用的拟三维地下水数值模型,模型由水平方向二维地下水数值模型、基于水平离散网格的垂直一维土壤水流模型和水平网格内两者的联系方程构成；同时采用自适应网格加密方法提出了一种新的土壤垂直分层结构来计算各层节点的土壤含水量；通过理想试验对模型的河水水位、地表水力传导率和地表通量进行敏感性分析以及结合SCE-UA参数优化方法,对塔里木河下游英苏断面上的地下水位进行了模拟验证,表明模型具有在不同条件下对地下水埋深的模拟能力；同时对塔里木河流域地下水位进行数值模拟试验,检验了地下水侧向流对大尺度起伏地形与河流水条件的响应,结果表明模型具有模拟大尺度地下水侧向流动的能力.
(3)发展了二维土壤水模型基于POD的降维数值解法.针对二维非饱和土壤水流方程问题,发展分别基于隐式差分格式和有限元格式的特征正交分解(POD)降维格式,它们具有计算量很少且足够高精度的特点,并给出其简化的POD隐式差分格式和POD有限元格式解的误差分析；数值例子表明在简化的POD隐式差分格式和POD有限元格式与通常的差分格式和有限元格式解之间的误差足够小的情况下,POD隐式差分格式和有限元格式比通常的差分格式和有限元格式大大地节省计算量,从而验证POD方法的有效性.
Soil water connecting surface water and groundwater affects climate by influenc-ing the water and energy exchange between land surface and atmosphere, and it plays a important role in climate and water circulation system. Dynamic variations in the water table over a region directly influence soil moisture at the surface, which leads to effects on latent and sensible heat fluxes and the growth and development of natural vegetation over the region. Water table fluctuations are influenced by surface fluxes such as infil-tration or evapotranspiration through the unsaturated zone, thus there is an important connection between soil water and ground water.
It is difficult to represent reasonably the interaction between soil water and ground water in climate and land circulation model with the fully three-dimensional soil wa-ter and groundwater model. There are many computational quantities when the three-dimensional interaction model is applied to large-scale regions. According to the main character of soil water flow and ground water flow in land surface processes, we have developed a quasi three-dimensional groundwater model including the equations of one-dimensional vertical soil water flow and two-dimensional horizonal groundwater flow. The model reduces computation and make it possible that representing reasonably the interaction of soil water and groundwater in climate and land water circulation model.
The numerical computation is the most effective method in solving the two-dimensi onal or high-dimensional partial difference equation. however, the discrete equations ob-tained by numerical method tend to be more unknown variables, the computation will be time-consuming. Proper orthogonal decomposition (POD) is an effective method for ap-proximating a large amount of data. Reducing computational dimensions、diminishing computational quantity and saving computation time are the most obvious advantage of POD. Combining the POD and usually numerical computation method and constructing a kind of numeric format with high accuracy and lower free degrees will be the broad application prospects.
Research objective of this paper is building a quasi three-dimensional ground-water model suitable for the simulation of the climate and land water circulation system, and developing a kind of dimension-reduction numerical solution based on POD for the two-dimensional soil water model, so as to improve numerical compu- tation efficiency. The main content of this paper includes as follows:
(1) Built a quasi two-dimensional numerical model based on soil water-groundwater and stream-aquifer interactions. The estimation of water table depth and soil water content on the soil profile under stream-aquifer interaction was reduced to be the two-dimensional moving boundary problem with soil water-groundwater interaction. The soil water flow was approximately divided into an unsaturated vertical soil water flow and a horizontal groundwater flow for the simulation of the interaction between soil water and groundwater. We developed a quasi two-dimensional numeric model based on vertical soil water equation and horizontal water table equation under stream-aquifer interaction. The sensitivities of the main parameters of the developed model were dis-cussed and its validation at the Yingsu Section of the lower reaches of the Tarim River with stream water conveyance showed the model can simulate water table dynamic vari-ation reasonably.
(2) Built a quasi three-dimensional variably saturated groundwater numeri-cal model based on soil water-groundwater interaction. The interaction of soil water and groundwater was reduced to be a three-dimensional saturated-unsaturated soil water flows problem. We developed a quasi three-dimensional, variably saturated groundwater flow model by approximately dividing the three-dimensional soil water and groundwater flow into an unsaturated vertical soil water flow and a horizontal groundwater flow for simulation of the interactions between soil water and groundwater. For each unsaturated soil column, a new soil layer structure was presented and an adaptive grid refinement method was applied to simulate the soil moisture content at the cell nodes. Synthetic ex-periments using the quasi three-dimensional numerical model were conducted to test the sensitivities of the main parameters of the model, including the river elevation, ground surface horizontal hydraulic conductivity, and surface flux. The results showed the ro-bustness of the developed model under different conditions. The estimation scheme of water table depth based the developed quasi three-dimensional model and SCE-UA pa-rameter calibration method was validated by a case of stream water conveyance in the lower reaches of the Tarim River. The simulation results of the water table elevations were in good agreement with the observations in the four monitoring wells. The param-eter test and model application for the prediction of water table elevations at the Yingsu section in the lower reaches of the Tarim River showed that it can be more reasonable to use the developed model with an adaptive refinement grid to substitute the original ten-layer unsaturated soil water structure. A numerical experiment by the model for Tarim River Basin was conducted to test the model on simulation of the effects of lateral groundwater flow for large-scale high-relief topography with stream water conveyance. The results showed that the model had the potential to be used in large-scale groundwa-ter latent flow.
(3)Developed dimension-reduce numerical method based on POD for the two-dimensional unsaturated soil water flow equation. For the two-dimensional unsatu-rated soil water equation, an optimizing reduced implicit difference scheme (IDS) and reduced finite element scheme (FES) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) was presented, respectively. The error esti-mates of the corresponding reduced POD numerical scheme were given and proved. The numerical examples showed the error between the reduced POD solution and usu-ally numerical solution was small enough, and computation time was reduced. Thus, both the feasibility and efficiency of our reduced IDS were validated.
引文
[1]雷志栋,杨诗秀,谢森传,土壤水动力学,清华大学出版社,(1987).
[2]沈照理,刘光亚,杨成田,水文地质学,科学出版社,(1982).
[3]薛禹群,地下水动力学(第二版),地质出版社,(1997).
[4]谢正辉,曾庆存,戴永久,王斌有限元集中质量法在非饱和土壤水流中的应用,气候与环境研究,3(1998)73-81.
[5]谢正辉,曾庆存,戴永久,等,非饱和流问题的数值模拟研究,中国科学D辑：地球科学,28(1998)175-280.
[6]罗振东,谢正辉,朱江,非饱和水流问题的混合元法及其数值模拟,计算数学,25(2003)113-128.
[7]谢正辉,罗振东,曾庆存,等,非饱和土壤水流问题含水量和通量的数值模拟研究,自然科学进展,4(1999)1280-1286.
[8]李焕荣,罗振东,谢正辉,朱江,非饱和土壤水流问题的广义差分法及其数值模拟,计算数学,28(2006)321-336.
[9]X. Chen, Q. Hu, Groundwater influences on soil moisture and surface evaporation, J. Hy-drol.,297 (2004) 285-300.
[10]R.L. Walko and Coauthors, Coupled atmosphere-biophysics-hydrology models for envi-ronmental modeling, J. Appl. Meteorol.,39 (2000) 931-944.
[11]M. Sophocleous, Interactions between groundwater and surface water:the state of the science, Hydrogeol. J.,10 (2002) 52-67.
[12]J.P. York, M. Person, WJ. Gutowski and T.C. Winter, Putting aquifers into atmospheric simulation models:an example from the Mill Creek Watershed, northeastern Kansas, Adv. Water Resour.,25 (2002) 221-238.
[13]G. Seuffert, P. Gross, C. Simmer and E. Wood, The influence of hydrologic modeling on the predicted local weather:Two-way coupling of a mesoscale weather prediction model and a land surface hydrologic model, J. Hydrometeorol.,3 (2002) 505-523.
[14]J.B. Levine, G.D. Salvucci, Equilibrium analysis of groundwatervadose zone interactions and the resulting spatial distribution of hydrologic fluxes across a Canadian prairie, Water Resour. Res.,35 (1999) 1369-1383.
[15]J. Chen, P. Kumar, Topographic influence on the seasonal and interannual variation of water and energy balance of basins in North America, J. Hydrometeorol.,14 (2001) 1989-2014.
[16]R. Koster, M. Suarez, A. Ducharne, M. Stieglitz and P. Kumar, A catchment-based ap-proach to modeling land surface processes in a general circulation model 1. Model struc-ture, J. Geophys. Res.,105 (2000) 24809-24822.
[17]P. Yeh, E. Eltahir, Representation of water table dynamics in a land surface scheme. Part I: Model development, J. Clim.,18 (2005) 1861-1880.
[18]P. Yeh, E. Eltahir, Representation of water table dynamics in a land surface scheme. Part II:Subgrid variability., J. Clim.,18 (2005) 1881-1901.
[19]N. Gedney, P. Cox, The sensitivity of global climate model simulations to the representa-tion of soil moisture heterogeneity, J. Hydrometeorol.,4 (2003) 1265-1275.
[20]G.Y. Niu, Z.L. Yang, R.E. Dickinson, L.E. Gulden and H. Su, Development of a simple groundwater model for use in climate models and evaluation with Grav-ity Recovery and Climate Experiment data, J. Geophys. Res.,112 (2007) D07103, doi:10.1029/2006JD007522.
[21]X. Yuan, Z.H. Xie and M.L. Liang, Spatiotemporal prediction of shallow water table depths in continental China, Water Resour. Res.,44 (2008) W04414, doi: 10.1029/2006WR005453.
[22]陈亚宁,张小雷,祝向民,等,新疆塔里木河下游断流河道输水的生态效应分析,中国科学D辑：地球科学,34(2004)475-482.
[23]Z.H. Xie, X. Yuan, Prediction of water table under stream-aquifer interactions over an arid region, Hydrol. Process.,24 (2009) 160-169.
[24]X.J. Tian, Z.H. Xie, S.L. Zhang and M.L. Liang, A subsurface runoff parameterization with water storage and recharge based on the Boussinesq-storage equation for a land surface model, Sci. China Ser. D-Earth Sci.,49 (2006) 622-631.
[25]V.Y. Ivanov, E.R. Vivoni, R.L. Bras and D. Entekhabi, Preserving high-resolution surface and rainfall data in operational-scale basin hydrology:A fully-distributed physicallybased approach, J. Hydrol.,298 (2004) 80-111.
[26]V.Y. Ivanov, E.R. Vivoni, R.L. Bras and D. Entekhabi, Catchment hydrologic response with a fully distributed triangulated irregular network model, Water Resour. Res.,40 (2004) W11102, doi:10.1029/2004WR 003218.
[27]E.R. Vivoni, D. Entekhabi, R.L. Bras and V.Y. Ivanov, Controls on runoff generation and scale-dependence in a distributed hydrologic model, Hydrol. Earth Syst. Sci.,4 (2007) 983-1029.
[28]S J. Kollet, R.M. Maxwell, Integrated surface-groundwater flow modeling:A free-surface overland flow boundary condition in a parallel groundwater flow model, Adv. Water Re-sour.,29 (2006) 945-958.
[29]S.J. Kollet, R.M. Maxwell, Capturing the influence of groundwater dynamics on land sur-face processes using an integrated, distributed watershed model, Water Resour. Res.,44 (2008) W02402, doi:10.1029/2007WR006004.
[30]R.M. Maxwell, F.K. Chow and S.J. Kollet, The groundwater-land-surface-atmosphere con-nection:Soil moisture effects on the atmospheric boundary layer in fully-coupled simula-tions, Adv. Water Resour.,30 (2007) 2447-2466.
[31]K. Christiaens, J. Feyen, Use of sensitivity and uncertainty measures in distributed hydro-logical modeling with an application to the MIKE SHE model, Water Resour. Res.,38 (2002) 1169, doi:10.1029/2001 WR000478.
[32]X. Liang, Z.H. Xie and M.Y. Huang, A new parameterization for surface and groundwater interactions and its impact on water budgets with the variable infiltration capacity (VIC) land surface model, J. Geophys. Res.,108 (2003) 8613, doi:10.1029/2002JD003090.
[33]X. liang, Z.H. Xie, Important factors in land-atmosphere interactions:Surface runoff gen-erations and interactions between surface and groundwater, Glob Planet Change,38 (2003) 101-114.
[34]杨宏伟,谢正辉,陆面模式VIC中动态表示地下水位的新方法,自然科学进展,13(2003)615-620.
[35]X. Yuan, Z.H. Xie, J. Zheng, X J. Tian and Z.L. Yang, Effects of water table dynamics on regional climate:A case study over east Asian monsoon area, J.Geophys. Res.,113 (2008) D21112, doi:10.1029/2008JD010180.
[36]冯康,基于变分原理的差分格式,应用数学与计算数学,2(1965)238-262.
[37]李荣华,两点边值问题的广义差分法,吉林大学自然科学学报,1(1982)26-40.
[38]李荣华,陈仲英,微分方程的广义差分法,吉林大学出版社,(1994).
[39]Q. Li, Generalized difference method, Lecture Notes of the Twelfth Mathematical Work-shop, Taejon, Korea, (1997).
[40]P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, Coherent Structures,Dynamical Sys-tems and Symmetry, Cambridge University Press, (1996).
[41]K. Fukunaga, Introduction to Statistical Recognition, Academic Press, (1990).
[42]I.T. Jolliffe, Principal Component Analysis, Springer-Verlag, (2002).
[43]D.T. Crommelin and A.J. Majda, Strategies for model reduction:comparing different op-timal Bases, J. Atmos. Sci.,61 (2004) 2306-2317.
[44]A.J. Majda, I. Timofeyev and E. Vanden-Eijnden, Systematic strategies for stochastic mode reduction in climate, J. Atmos. Sci.,60 (2003) 1705-1723.
[45]F. Selten, Baroclinic empirical orthogonal functions as basis functions in an atmospheric Model, J. Atmos. Sci.,54 (1997) 2100-2114.
[46]P. Moin, R. Moser, Characteristic-eddy decomposition of turbulence in channel, J. Fluid Mech.,200(1989)471-509.
[47]M. Rajaee, S. Karlsson and L. Sirovich, Low dimensional description of free shear flow coherent structures and their dynamical behavior, J. Fluid Mech.,258 (1994) 1-29.
[48]R. Joslin, M. Gunzburger, R. Nicolaides, G. Erlebacher and M. Hussaini, A self-contained automated methodology for optimal flow control validated for transition delay, AIAA Jour-nal,35 (1997) 816-824.
[49]H. Ly, H. Tran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, Q. Appl. Math.,60 (2002) 631-656.
[50]J. Lumley, Coherent Structures in Turbulence. In Transition and Turbulence, Meyer R E (ed.), New York Press, (1981).
[51]N. Aubry, P. Holmes, J. Lumley and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid D.,192 (1998) 115-173.
[52]L. Sirovich, Turbulence and the dynamics of coherent structures:Part Ⅰ-Ⅲ, Q. Appl. Math., 45(1987)561-590.
[53]K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math.,90 (2001) 117-148.
[54]K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal.,40 (2002) 492-515.
[55]Z.D. Luo, R.W. Wang and J. Zhu, Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations, Sci. China Ser A-Math.,50 (2007)1186-1196.
[56]Z.D. Luo, J. Chen, I.M. Navon and X.Z. Yang, Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations, SIAM J. Numeri. Analy.,47 (2008) 1-19.
[57]Z.D. Luo, J. Chen, P. Sun and X.Z. Yang, Finite element formulation based on proper orthogonal decomposition for parabolic equations, Sci. China Ser A-Math.,52 (2009) 587-596.
[58]Z.D. Luo, X.Z. Yang and Y.J. Zhou, A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation, J. Com-put. Appl. Math.,229 (2009) 97-107.
[59]Z.D. Luo, X.Z. Yang and Y.J. Zhou, A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation, Appl. Numer.l Math.,59 (2009) 1933-1946.
[60]H. Ly, H. Iran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, Q. Appl. Math.,60 (2002) 631-656.
[61]Y.H. Cao, J. Zhu, Z.D. Luo and I.M. Navon, Reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition, Comput. Math. Appl.,52 (2006)1373-1386.
[62]Y.H. Cao, J. Zhu, I.M. Navon and Z.D. Luo, A reduced order approach to four-dimensional variational data assimilation using proper orthogonal decomposition, Int. J. Numer. Meth. Fl.,53 (2007)1571-1583.
[63]Z.D. Luo, J. Zhu, R.W. Wang and I.M. Navon, Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model, Comput. Meth. Appl. Mech. En.,196 (2007) 4184-4195.
[64]Z.D. Luo, J. Chen, J. Zhu, R.W. Wang and I.M. Navon, An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model, Int. J. Numer. Meth. F.,55 (2007) 143-161.
[65]R.W. Wang, J. Zhu, Z.D. Luo and I.M. Navon, An equation-free reduced order modeling approach to tropic pacific simulation, Adv. Geosc,12 (2007) 1-16.
[66]J. Du, J. Zhu, Z.D. Luo and I.M. Navon, An optimizing finite difference scheme based on proper orthogonal decomposition for CVD equations, Commun. Numer. Meth. En., (2009) DOI:10.1002/cnm.1295.
[67]H. Yu, H. Chu, Understanding space-time patterns of groundwater system by empirical orthogonal functions:A case study in the Choshui River alluvial fan, Taiwan, J. Hydrol., 381 (2010)239-247.
[68]T. Bernard, O. Krol, F. IITB, H. Linke and T. Rauschenbach, Optimal Management of Regional Water Supply Systems Using a Reduced Finite-Element Groundwater Model, Automatisierungstechnik, (2009) DOI 10.1524/auto.2009.0807.
[69]K.W. Oleson and Coauthors, Technical description of the Community Land Model (CLM), NCAR/TN-461+STR, (2004).
[70]J. Bear, Dynamics of Fluids in Porous Media, Dover Pub Inc, (1972).
[71]Q. Y. Duan, V. K. Gupta and S. Sorooshian, Effective and efficient global optimization for conceptual rainfall-runoff model, Water Resour. Res.,28 (1992) 1015-1031.
[72]Q.Y. Duan, V. K. Gupta and S. Sorooshian, A shuffled complex evolution approach for effective and efficient global minimization, J. Optim. Theory Appl.,76 (1993) 501-521.
[73]Q.Y. Duan, S. Sorooshian and V.K. Gupta, Optimal use of the SCE-UA global optimization method for calibrating watershed models, J. Hydrol.,15 (1994) 265-284.
[74]R.B. Clapp, G.M. Hornberger, Empirical equation for some soil hydraulic properties, Wa-ter Resour. Res.,14 (1978) 601-604.
[75]宋郁东,樊自立,雷志栋,等,中国塔里木河水资源与生态问题研究,新疆人民出版社,(2000).
[76]Q. Feng, G.D. Cheng, Towards sustainable development of the environmentally degraded arid rivers of China-a case study from Tarim River, Environ. Geol.,41 (2001) 229-238.
[77]陈亚宁,崔旺诚,李卫红,等,塔里木河的水资源利用与生态保护,地理学报,58(2003)215-222.
[78]徐海量,陈亚宁,李卫红,塔里木河下游生态输水后地下水的响应研究,环境科学研究,16(2003)19-28.
[79]陈亚宁,李卫红,陈亚鹏,等,新疆塔里木河下游断流河道输水与生态恢复,生态学报,27(2007)538-545.
[80]叶朝霞,陈亚宁,李卫红,塔里木河下游生态输水对地下水位影响的综合评价,干旱区资源与环境,2(2007)12-16.
[81]张丽华,陈亚宁,李卫红,塔里木河下游生态输水对植物群落数量特征的影响,干旱区研究,23(2006)32-38.
[82]湾疆辉,陈亚宁,李卫红,等,塔里木河下游断流河道输水后潜水埋深变化规律研究,干旱区地理,31(2008)428-435.
[83]杨玉海,陈亚宁,李卫红,塔里木河下游土壤特性及荒漠化程度研究,水土保持学报,21(2007)44-49.
[84]高艳红,彭红春,李海英,等,黑河流域土壤质地分类数据建立及其模效果检验,高原气象,26(2007)967-974.
[85]R.E. Dickison and Coauthors, Biosphere atmosphere transfer scheme (BATS) for NCAR community climate model, NCAR Technical Note, NCAR/TN 275+STR, (1986).
[86]Y.N. Chen, Y. P. Chen and C.C. Xu, Effects of ecological water conveyance on groundwa-ter dynamics and riparian vegetation in the lower reaches of Tarim River, China, Hydrol Process, (2009) doi:10.1002/hyp.7429.
[87]Y. Fan,, G. Miguez-Macho, C. Weaver, R. Walko and A. Robock, Incorporating water table dynamics in climate modeling:1. Water table observations and equilibrium water table simulations, J. Geophys. Res.,112 (2007) D10125, doi:10.1029/2006JD008111.
[88]张蔚榛,地下水与土壤水动力学,中国水利水电出版社,(1996).
[89]陈亚浙,二阶抛物型偏微分方程,北京大学出版社,(2003).
[90]罗振东,混合有限元方法基础及其应用,科学出版社,(2006).
[91]V. Girault, P.A. Raviart, Finite element methods for Navier-Stokes equation, Theorem and Algorithms, Springer-Verlag, (1986).
[92]V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer, (1997).
[93]P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, (1978).
[2]沈照理,刘光亚,杨成田,水文地质学,科学出版社,(1982).
[3]薛禹群,地下水动力学(第二版),地质出版社,(1997).
[4]谢正辉,曾庆存,戴永久,王斌有限元集中质量法在非饱和土壤水流中的应用,气候与环境研究,3(1998)73-81.
[5]谢正辉,曾庆存,戴永久,等,非饱和流问题的数值模拟研究,中国科学D辑：地球科学,28(1998)175-280.
[6]罗振东,谢正辉,朱江,非饱和水流问题的混合元法及其数值模拟,计算数学,25(2003)113-128.
[7]谢正辉,罗振东,曾庆存,等,非饱和土壤水流问题含水量和通量的数值模拟研究,自然科学进展,4(1999)1280-1286.
[8]李焕荣,罗振东,谢正辉,朱江,非饱和土壤水流问题的广义差分法及其数值模拟,计算数学,28(2006)321-336.
[9]X. Chen, Q. Hu, Groundwater influences on soil moisture and surface evaporation, J. Hy-drol.,297 (2004) 285-300.
[10]R.L. Walko and Coauthors, Coupled atmosphere-biophysics-hydrology models for envi-ronmental modeling, J. Appl. Meteorol.,39 (2000) 931-944.
[11]M. Sophocleous, Interactions between groundwater and surface water:the state of the science, Hydrogeol. J.,10 (2002) 52-67.
[12]J.P. York, M. Person, WJ. Gutowski and T.C. Winter, Putting aquifers into atmospheric simulation models:an example from the Mill Creek Watershed, northeastern Kansas, Adv. Water Resour.,25 (2002) 221-238.
[13]G. Seuffert, P. Gross, C. Simmer and E. Wood, The influence of hydrologic modeling on the predicted local weather:Two-way coupling of a mesoscale weather prediction model and a land surface hydrologic model, J. Hydrometeorol.,3 (2002) 505-523.
[14]J.B. Levine, G.D. Salvucci, Equilibrium analysis of groundwatervadose zone interactions and the resulting spatial distribution of hydrologic fluxes across a Canadian prairie, Water Resour. Res.,35 (1999) 1369-1383.
[15]J. Chen, P. Kumar, Topographic influence on the seasonal and interannual variation of water and energy balance of basins in North America, J. Hydrometeorol.,14 (2001) 1989-2014.
[16]R. Koster, M. Suarez, A. Ducharne, M. Stieglitz and P. Kumar, A catchment-based ap-proach to modeling land surface processes in a general circulation model 1. Model struc-ture, J. Geophys. Res.,105 (2000) 24809-24822.
[17]P. Yeh, E. Eltahir, Representation of water table dynamics in a land surface scheme. Part I: Model development, J. Clim.,18 (2005) 1861-1880.
[18]P. Yeh, E. Eltahir, Representation of water table dynamics in a land surface scheme. Part II:Subgrid variability., J. Clim.,18 (2005) 1881-1901.
[19]N. Gedney, P. Cox, The sensitivity of global climate model simulations to the representa-tion of soil moisture heterogeneity, J. Hydrometeorol.,4 (2003) 1265-1275.
[20]G.Y. Niu, Z.L. Yang, R.E. Dickinson, L.E. Gulden and H. Su, Development of a simple groundwater model for use in climate models and evaluation with Grav-ity Recovery and Climate Experiment data, J. Geophys. Res.,112 (2007) D07103, doi:10.1029/2006JD007522.
[21]X. Yuan, Z.H. Xie and M.L. Liang, Spatiotemporal prediction of shallow water table depths in continental China, Water Resour. Res.,44 (2008) W04414, doi: 10.1029/2006WR005453.
[22]陈亚宁,张小雷,祝向民,等,新疆塔里木河下游断流河道输水的生态效应分析,中国科学D辑：地球科学,34(2004)475-482.
[23]Z.H. Xie, X. Yuan, Prediction of water table under stream-aquifer interactions over an arid region, Hydrol. Process.,24 (2009) 160-169.
[24]X.J. Tian, Z.H. Xie, S.L. Zhang and M.L. Liang, A subsurface runoff parameterization with water storage and recharge based on the Boussinesq-storage equation for a land surface model, Sci. China Ser. D-Earth Sci.,49 (2006) 622-631.
[25]V.Y. Ivanov, E.R. Vivoni, R.L. Bras and D. Entekhabi, Preserving high-resolution surface and rainfall data in operational-scale basin hydrology:A fully-distributed physicallybased approach, J. Hydrol.,298 (2004) 80-111.
[26]V.Y. Ivanov, E.R. Vivoni, R.L. Bras and D. Entekhabi, Catchment hydrologic response with a fully distributed triangulated irregular network model, Water Resour. Res.,40 (2004) W11102, doi:10.1029/2004WR 003218.
[27]E.R. Vivoni, D. Entekhabi, R.L. Bras and V.Y. Ivanov, Controls on runoff generation and scale-dependence in a distributed hydrologic model, Hydrol. Earth Syst. Sci.,4 (2007) 983-1029.
[28]S J. Kollet, R.M. Maxwell, Integrated surface-groundwater flow modeling:A free-surface overland flow boundary condition in a parallel groundwater flow model, Adv. Water Re-sour.,29 (2006) 945-958.
[29]S.J. Kollet, R.M. Maxwell, Capturing the influence of groundwater dynamics on land sur-face processes using an integrated, distributed watershed model, Water Resour. Res.,44 (2008) W02402, doi:10.1029/2007WR006004.
[30]R.M. Maxwell, F.K. Chow and S.J. Kollet, The groundwater-land-surface-atmosphere con-nection:Soil moisture effects on the atmospheric boundary layer in fully-coupled simula-tions, Adv. Water Resour.,30 (2007) 2447-2466.
[31]K. Christiaens, J. Feyen, Use of sensitivity and uncertainty measures in distributed hydro-logical modeling with an application to the MIKE SHE model, Water Resour. Res.,38 (2002) 1169, doi:10.1029/2001 WR000478.
[32]X. Liang, Z.H. Xie and M.Y. Huang, A new parameterization for surface and groundwater interactions and its impact on water budgets with the variable infiltration capacity (VIC) land surface model, J. Geophys. Res.,108 (2003) 8613, doi:10.1029/2002JD003090.
[33]X. liang, Z.H. Xie, Important factors in land-atmosphere interactions:Surface runoff gen-erations and interactions between surface and groundwater, Glob Planet Change,38 (2003) 101-114.
[34]杨宏伟,谢正辉,陆面模式VIC中动态表示地下水位的新方法,自然科学进展,13(2003)615-620.
[35]X. Yuan, Z.H. Xie, J. Zheng, X J. Tian and Z.L. Yang, Effects of water table dynamics on regional climate:A case study over east Asian monsoon area, J.Geophys. Res.,113 (2008) D21112, doi:10.1029/2008JD010180.
[36]冯康,基于变分原理的差分格式,应用数学与计算数学,2(1965)238-262.
[37]李荣华,两点边值问题的广义差分法,吉林大学自然科学学报,1(1982)26-40.
[38]李荣华,陈仲英,微分方程的广义差分法,吉林大学出版社,(1994).
[39]Q. Li, Generalized difference method, Lecture Notes of the Twelfth Mathematical Work-shop, Taejon, Korea, (1997).
[40]P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, Coherent Structures,Dynamical Sys-tems and Symmetry, Cambridge University Press, (1996).
[41]K. Fukunaga, Introduction to Statistical Recognition, Academic Press, (1990).
[42]I.T. Jolliffe, Principal Component Analysis, Springer-Verlag, (2002).
[43]D.T. Crommelin and A.J. Majda, Strategies for model reduction:comparing different op-timal Bases, J. Atmos. Sci.,61 (2004) 2306-2317.
[44]A.J. Majda, I. Timofeyev and E. Vanden-Eijnden, Systematic strategies for stochastic mode reduction in climate, J. Atmos. Sci.,60 (2003) 1705-1723.
[45]F. Selten, Baroclinic empirical orthogonal functions as basis functions in an atmospheric Model, J. Atmos. Sci.,54 (1997) 2100-2114.
[46]P. Moin, R. Moser, Characteristic-eddy decomposition of turbulence in channel, J. Fluid Mech.,200(1989)471-509.
[47]M. Rajaee, S. Karlsson and L. Sirovich, Low dimensional description of free shear flow coherent structures and their dynamical behavior, J. Fluid Mech.,258 (1994) 1-29.
[48]R. Joslin, M. Gunzburger, R. Nicolaides, G. Erlebacher and M. Hussaini, A self-contained automated methodology for optimal flow control validated for transition delay, AIAA Jour-nal,35 (1997) 816-824.
[49]H. Ly, H. Tran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, Q. Appl. Math.,60 (2002) 631-656.
[50]J. Lumley, Coherent Structures in Turbulence. In Transition and Turbulence, Meyer R E (ed.), New York Press, (1981).
[51]N. Aubry, P. Holmes, J. Lumley and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid D.,192 (1998) 115-173.
[52]L. Sirovich, Turbulence and the dynamics of coherent structures:Part Ⅰ-Ⅲ, Q. Appl. Math., 45(1987)561-590.
[53]K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math.,90 (2001) 117-148.
[54]K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal.,40 (2002) 492-515.
[55]Z.D. Luo, R.W. Wang and J. Zhu, Finite difference scheme based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations, Sci. China Ser A-Math.,50 (2007)1186-1196.
[56]Z.D. Luo, J. Chen, I.M. Navon and X.Z. Yang, Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the non-stationary Navier-Stokes equations, SIAM J. Numeri. Analy.,47 (2008) 1-19.
[57]Z.D. Luo, J. Chen, P. Sun and X.Z. Yang, Finite element formulation based on proper orthogonal decomposition for parabolic equations, Sci. China Ser A-Math.,52 (2009) 587-596.
[58]Z.D. Luo, X.Z. Yang and Y.J. Zhou, A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation, J. Com-put. Appl. Math.,229 (2009) 97-107.
[59]Z.D. Luo, X.Z. Yang and Y.J. Zhou, A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation, Appl. Numer.l Math.,59 (2009) 1933-1946.
[60]H. Ly, H. Iran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, Q. Appl. Math.,60 (2002) 631-656.
[61]Y.H. Cao, J. Zhu, Z.D. Luo and I.M. Navon, Reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition, Comput. Math. Appl.,52 (2006)1373-1386.
[62]Y.H. Cao, J. Zhu, I.M. Navon and Z.D. Luo, A reduced order approach to four-dimensional variational data assimilation using proper orthogonal decomposition, Int. J. Numer. Meth. Fl.,53 (2007)1571-1583.
[63]Z.D. Luo, J. Zhu, R.W. Wang and I.M. Navon, Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model, Comput. Meth. Appl. Mech. En.,196 (2007) 4184-4195.
[64]Z.D. Luo, J. Chen, J. Zhu, R.W. Wang and I.M. Navon, An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model, Int. J. Numer. Meth. F.,55 (2007) 143-161.
[65]R.W. Wang, J. Zhu, Z.D. Luo and I.M. Navon, An equation-free reduced order modeling approach to tropic pacific simulation, Adv. Geosc,12 (2007) 1-16.
[66]J. Du, J. Zhu, Z.D. Luo and I.M. Navon, An optimizing finite difference scheme based on proper orthogonal decomposition for CVD equations, Commun. Numer. Meth. En., (2009) DOI:10.1002/cnm.1295.
[67]H. Yu, H. Chu, Understanding space-time patterns of groundwater system by empirical orthogonal functions:A case study in the Choshui River alluvial fan, Taiwan, J. Hydrol., 381 (2010)239-247.
[68]T. Bernard, O. Krol, F. IITB, H. Linke and T. Rauschenbach, Optimal Management of Regional Water Supply Systems Using a Reduced Finite-Element Groundwater Model, Automatisierungstechnik, (2009) DOI 10.1524/auto.2009.0807.
[69]K.W. Oleson and Coauthors, Technical description of the Community Land Model (CLM), NCAR/TN-461+STR, (2004).
[70]J. Bear, Dynamics of Fluids in Porous Media, Dover Pub Inc, (1972).
[71]Q. Y. Duan, V. K. Gupta and S. Sorooshian, Effective and efficient global optimization for conceptual rainfall-runoff model, Water Resour. Res.,28 (1992) 1015-1031.
[72]Q.Y. Duan, V. K. Gupta and S. Sorooshian, A shuffled complex evolution approach for effective and efficient global minimization, J. Optim. Theory Appl.,76 (1993) 501-521.
[73]Q.Y. Duan, S. Sorooshian and V.K. Gupta, Optimal use of the SCE-UA global optimization method for calibrating watershed models, J. Hydrol.,15 (1994) 265-284.
[74]R.B. Clapp, G.M. Hornberger, Empirical equation for some soil hydraulic properties, Wa-ter Resour. Res.,14 (1978) 601-604.
[75]宋郁东,樊自立,雷志栋,等,中国塔里木河水资源与生态问题研究,新疆人民出版社,(2000).
[76]Q. Feng, G.D. Cheng, Towards sustainable development of the environmentally degraded arid rivers of China-a case study from Tarim River, Environ. Geol.,41 (2001) 229-238.
[77]陈亚宁,崔旺诚,李卫红,等,塔里木河的水资源利用与生态保护,地理学报,58(2003)215-222.
[78]徐海量,陈亚宁,李卫红,塔里木河下游生态输水后地下水的响应研究,环境科学研究,16(2003)19-28.
[79]陈亚宁,李卫红,陈亚鹏,等,新疆塔里木河下游断流河道输水与生态恢复,生态学报,27(2007)538-545.
[80]叶朝霞,陈亚宁,李卫红,塔里木河下游生态输水对地下水位影响的综合评价,干旱区资源与环境,2(2007)12-16.
[81]张丽华,陈亚宁,李卫红,塔里木河下游生态输水对植物群落数量特征的影响,干旱区研究,23(2006)32-38.
[82]湾疆辉,陈亚宁,李卫红,等,塔里木河下游断流河道输水后潜水埋深变化规律研究,干旱区地理,31(2008)428-435.
[83]杨玉海,陈亚宁,李卫红,塔里木河下游土壤特性及荒漠化程度研究,水土保持学报,21(2007)44-49.
[84]高艳红,彭红春,李海英,等,黑河流域土壤质地分类数据建立及其模效果检验,高原气象,26(2007)967-974.
[85]R.E. Dickison and Coauthors, Biosphere atmosphere transfer scheme (BATS) for NCAR community climate model, NCAR Technical Note, NCAR/TN 275+STR, (1986).
[86]Y.N. Chen, Y. P. Chen and C.C. Xu, Effects of ecological water conveyance on groundwa-ter dynamics and riparian vegetation in the lower reaches of Tarim River, China, Hydrol Process, (2009) doi:10.1002/hyp.7429.
[87]Y. Fan,, G. Miguez-Macho, C. Weaver, R. Walko and A. Robock, Incorporating water table dynamics in climate modeling:1. Water table observations and equilibrium water table simulations, J. Geophys. Res.,112 (2007) D10125, doi:10.1029/2006JD008111.
[88]张蔚榛,地下水与土壤水动力学,中国水利水电出版社,(1996).
[89]陈亚浙,二阶抛物型偏微分方程,北京大学出版社,(2003).
[90]罗振东,混合有限元方法基础及其应用,科学出版社,(2006).
[91]V. Girault, P.A. Raviart, Finite element methods for Navier-Stokes equation, Theorem and Algorithms, Springer-Verlag, (1986).
[92]V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer, (1997).
[93]P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, (1978).