各向异性外问题的Schwarz交替法及其收敛性和误差估计  

A SCHWARZ ALTERNATING METHOD AND ITS CONVERGENCE AND ERROR ESTIMATE FOR ANISOTROPIC ELLIPTIC EXTERIOR PROBLEMS

作　　者：郑权[1] 白荣霞[1] 董俊雨[1] 

机构地区：[1]北方工业大学理学院,北京100144

出　　处：《计算数学》2009年第1期65-76,共12页Mathematica Numerica Sinica

基　　金：北京市自然科学基金(1072009)资助项目.

摘　　要：本文对于无界区域各向异性常系数椭圆型偏微分方程研究了一种基于自然边界归化的Schwarz交替法.利用极值原理证明了在连续情形最大模意义下的几何迭代收敛性,通过选取适当的共焦椭圆边界利用Fourier分析获得了不依赖各向异性程度的最优的迭代收缩因子.还在离散情形最大模意义下证明了几何收敛性,而且进一步得到了误差估计.最后,数值结果证实了迭代收缩因子和误差估计的正确性,表明了该方法在无界区域上求解各向异性椭圆型偏微分方程的优越性.We investigate a Schwarz alternating method based on the natural boundary reduction on the elliptic boundary for the anisotropic elliptic PDEs with constant coefficients in unbounded domains. We prove its geometric iterative convergence with maximum norm in the continuous case by using the maximum principle, and obtain an optimal iteration contract factor, which is independent of the anisotropic degree, by using Fourier analysis with confocal elliptic boundaries. We also prove its geometric convergence in the discrete case with maximum norm and obtain an error estimate of the iterative convergent solution. Finally, our numerical results confirm the correctness of the iterative contract factor and the error estimate, and show the advantage of this method for solving the anisotropic elliptic PDEs in unbounded domains.

关 键 词：SCHWARZ交替法 无界区域 各向异性问题 自然边界归化 误差估计 

分 类 号：O241.82[理学—计算数学] O151.21[理学—数学]