区域分解预处理器研究及其在地下水模拟中的应用  

Domain decomposition preprocessor research and its application in groundwater modeling

作　　者：王佩[1] 朱国荣[1] 江思珉[2] 孙斌堂 

机构地区：[1]南京大学水科学系,南京210093 [2]同济大学水利工程系,上海200092 [3]中国冶金地质工程勘察总局山东局,济南250014

出　　处：《南京大学学报（自然科学版）》2012年第6期753-760,共8页Journal of Nanjing University（Natural Science）

基　　金：国家自然科学基金(41002078)

摘　　要：地下水数值计算中,如何提高大型稀疏线性代数方程组的求解效率一直备受关注.条件预处理共轭梯度方法是求解大型稀疏线性代数方程组的有效方法,而如何构造高效的预处理器至关重要.本文基于有限元理论介绍了区域分解预处理器(domain decomposition preprocessor,DDP)的设计原理及实现过程,将其与预处理共轭梯度法结合为区域分解预处理共轭梯度法(DDP-PCG)并应用于地下水模拟中.文中首先采用具有解析解的均质稳定流模型验证了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,DDP-PCG求解,结果表明DDP-PCG迭代收敛次数几乎不随网格规模发生变化,具有很强的鲁棒性;随着网格规模的增加,DDP-PCG求解效率优势更加显著.最后针对承压含水层介质参数连续变化和突变两种非均质问题,对研究区进行大规模网格剖分,进一步证明在相同误差限下,DDP-PCG的求解效率高于其它三种方法.The problem that how to improve the efficiency of solving the large sparse linear algebraic equations has always been of concern in groundwater numerical simulation. Preconditioned conjugate gradient method is an effective method for solving large sparse linear algebraic equations. And how to construct efficient preprocessor is es- sential. The domain decomposition preprocessor design principles and implementation process are descirbed based on the finite element theory. The preconditioned conjugate gradient method with domain decomposition preconditioner （DDP-PCG） is proposed and applied to groundwater simulation. Firstly, DDP-PCG＇s reliability is proved with analytical homogeneous model. Then CG, Jacobi PCG, SSOR-PCG, DDP PCG are used to solve the model under different subdivisions,which show that DDP-PCG is of strong robustness and the higher the subdivision is,the more notable is DDP PCG＇s efficiency. Finally,inhomogeneous confined aquifer with continuous coefficients and abrupt coefficients are caculated under finely subdivision, which further prove that under the same error limit,DDP-PCG＇s efficiency are the highest of all.

关 键 词：区域分解预处理器 预处理共轭梯度法 有限元方法 地下水模拟 

分 类 号：O241.82[理学—计算数学]