求解Maxwell线性元鞍点系统的基于HX预条件子的Uzawa算法  

UZAWA ALGORITHM BASED ON HX PRECONDITIONER FOR SOLVING MAXWELL SADDLE-POINT SYSTEM

作　　者：王俊仙[1] 胡齐芽[2] 舒适[1] 

机构地区：[1]湘潭大学数学与计算科学学院,湖南湘潭411105 [2]中国科学院数学与系统科学研究院,北京100080

出　　处：《数值计算与计算机应用》2009年第4期305-314,共10页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金资助项目(10771178);高性能科学计算研究资助项目(2005CB321702);国家自然科学基金重点项目(G10531080);教育部重点项目(208093);湖南省研究生科研创新项目(CX2009B121)

摘　　要：首先对含跳系数的H^1型和H(curl)型椭圆问题的线性有限元方程,分别设计了基于AMG预条件子和基于节点辅助空间预条件子(HX预条件子)的PCG法.数值实验表明,算法的迭代次数基本不依赖于系数跳幅和离散网格"尺寸".然后以此为基础,对Maxwell方程组鞍点问题的第一类Nedelec线性棱元离散系统设计并分析了一种基于HX预条件子的Uzawa算法.当系数光滑时,理论上证明了算法的收敛率与网格规模无关.数值实验表明,新算法对跳系数情形也是高效和稳定的.We design two preconditioners for the linear finite element discrete system of H^1 and H（curl） elliptic problems with jump coefficients based on AMG and HX auxiliary spaces method,respectively.Numerical experiments indicate that the number of iterations is hardly dependent on mesh size and the jump coefficients.And then,we design and analyze a preconditioned Uzawa algorithm for solving the saddle-point system generated by the lowest order edge element discretization of Maxwell equations.In the Uzawa algorithm, we use HX preconditioner for the primal variable and AMG preconditioner for Schur complement. The theoretical analysis prove that the convergence rate is independent of mesh size if the coefficient is smooth. Furthermore, the numerical results also show that the new algorithm is efficient and robust for jump coefficients.

关 键 词：节点辅助空间预条件子 鞍点问题 UZAWA算法 跳系数 收敛率 

分 类 号：O241.82[理学—计算数学]