计算数学
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计算数学  2012, Vol. 34 Issue (1): 1-24    DOI:
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特征值问题的预变换方法 (Ⅱ):任意三角形域Laplace特征值的计算分析
孙家昶
中国科学院软件研究所并行计算实验室, 北京 100190
PRE-TRANSFORMED METHODS FOR EIGEN-PROBLEMS Ⅱ: EIGEN-STRUCTURE FOR LAPLACE EIGEN-PROBLEM OVER ARBITRARY TRIANGLES
Sun Jiachang
Laboratory of Parallel Software and Computational Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
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摘要 本文基于三类特殊三角形(等边、等腰直角及(30°,60°,90°)三角形域)Laplace特征函数系的构造,提出任意三角形区域上Laplace特征值的近似公式与算法.给出任意三角形域上所有特征值的逼近公式:λm,n≈π2/24S2(h12(7m2-12mn+7n2)+h22(3m2-4mn+3n2)-2h32(m2-4mn+n2)),m > n ≥1,特别, 对于最小特征值λmin2,1≈π2/S2 11h12+7h22+6h32/24,其中S是该三角形(h1h2h3)的面积,可作为数值PDE中三角剖分质量的一种新标准q(T):=3h32/16S2 11h12+7h22+6h32/24.结合数值计算与符号计算, 将这三类三角形的基底综合形成统一的新基底, 以反映几何(三条边)对于特征问题的影响, 从而提高任意三角形域的求解精度.
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关键词特征值问题的预变换方法   Laplace 特征值问题   任意三角形域     
Abstract: Based on Laplace eigen-structure over three special triangle domains (regular triangle, isoceles triangle and triangle with (30°, 60°, 90°)), we propose a unified basis to compute all Laplace eigenvalues over an arbitrary triangle with mixed numerical and symbolic computation. And a class of approximate formulas for evaluating all eigenvalues over an arbitrary triangle as λm,n≈π2/24S2(h12(7m2-12mn+7n2)+h22(3m2-4mn+3n2)-2h32(m2-4mn+n2)),Especially, for the smallest eigenvalue λmin≈π2/S2 11h12+7h22+6h32/24,where S is the area of the triangle with three lengths h1h2h3.And it can be as a new quality of 2-D triangle grid for 2-nd PDE problems as q(T):=3h32/16S2 11h12+7h22+6h32/24.To reflect the influence of the three side-lengths on the eigenvalues over an arbitrary triangle, we put the above three basis together and use numerical computation with some symbolic. This hybrid algorithm may a way to raise the accuracy of eigenvalues in computing.
Key wordsPre-transformed eigenvalues   eigen-vectors   Laplace PDE eigen-problem   Arbitrary triangle   
收稿日期: 2011-09-15;
基金资助:

国家自然基金项目资助 (No. 60970089, 61170075).

引用本文:   
. 特征值问题的预变换方法 (Ⅱ):任意三角形域Laplace特征值的计算分析[J]. 计算数学, 2012, 34(1): 1-24.
. PRE-TRANSFORMED METHODS FOR EIGEN-PROBLEMS Ⅱ: EIGEN-STRUCTURE FOR LAPLACE EIGEN-PROBLEM OVER ARBITRARY TRIANGLES[J]. Mathematica Numerica Sinica, 2012, 34(1): 1-24.
 
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