Effective condition number of finite difference method for poisson's equation involving boundary singularities
Effective condition number of finite difference method for poisson's equation involving boundary singularities
复制标题
涉及边界奇点的泊松方程有限差分法的有效条件数
DOI：
10.1080/01630563.2011.572267
复制
发表时间：
2012
影响因子：
1.2
通讯作者：
Tzon-Tzer Lu and Qing Fang
中科院分区：
文献类型：
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作者：
Zi-Cai Li;Hung-Tsai Huang;Chien-Sen Huang;Tzon-Tzer Lu and Qing Fang
为了求解具有对称正定矩阵A的线性代数方程组Ax = B，在[，]中，根据Chan，Foulser和Rice定义了有效条件数Cond_eff。Cond_eff比传统的条件数Cond小，或者小得多。此外，最简单的条件数Cond_EE也在[，]中定义。本文利用局部网格加密的有限差分方法研究了一种常见的含边界奇异性的泊松方程模型。导出了Cond_EE的界，从理论上表明有效条件数明显小于Cond。在本文中，通过探索局部加细性质，我们得到了最大步长h的有效条件数的界为O（1）和至少（h-1/2）。与在[，]中建立的边界O（h−3/2）相比，它们是显著的改进。因此，本文对有效条件数的研究达到了一个新的全面和先进的水平。
For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A, the effective condition number Cond_eff is defined in [, ] by following Chan and Foulser and Rice . The Cond_eff is smaller, or much smaller, than the traditional condition number Cond. Besides, the simplest condition number Cond_EE is also defined in [, ]. This article studies a popular model of Poisson's equation involving the boundary singularities by the finite difference method using the local refinements of grids. The bounds of Cond_EE are derived to display theoretically that the effective condition number is significantly smaller than the Cond. In this article, by exploring local refinement properties, we derive the bounds of effective condition numbers up toO(1) and at leasto(h−1/2) for the maximal step sizeh. They are significant improvements compared with the boundO(h−3/2), which is established in [, ]. Therefore, the study of effective condition number in this article reaches a new comprehensive and advanced level.