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Subject: Mathematics
Title:
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
Alternative Title: PRE-TRANSFORMED METHODS FOR EIGEN-PROBLEMSⅡ:EIGEN-STRUCTURE FOR LAPLACE EIGEN-PROBLEM OVER ARBITRARY TRIANGLES
Author: 孙家昶
Keyword: Pre-transformed eigenvalues ; eigen-vectors ; Laplace PDE eigen-problem ; Arbitrary triangle
Source: Mathematica Numerica Sinica
Issued Date: 2012
Volume: 34, Issue:1, Pages:40932
Department: 孙家昶, 中国科学院软件研究所并行计算实验室, 北京 100190, 中国.
Abstract: 本文基于三类特殊三角形(等边、等腰直角及(30degrees,60degrees,90degrees)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:lambdam,npi2/24S2(h_1~2 (7m~2-12mn+7n~2)+h_2~2(3m~2-4mn+3n~2)-2h_3~2(m~2-4mn+n~2)),(m>n≥1),特别,对于 最小特征值lambda_(min)=lambda_(2,1)pi~2/S~2 11h_1~2+7h_2~2+6h_3~2/24,其中S是该三角形(h_1≤h_2≤h_3)的面积,可作为数值PDE中三角剖分质量的一种新标准q (T):=3h_3~2/16S~2 11h~1_2+7h~2_2+6h_3~2/24.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.
English Abstract: Based on Laplace eigen-structure over three special triangle domains(regular triangle, isoceles triangle and triangle with(30degrees,60degrees,90degrees)),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 lambda_(m,n)pi~2/(24S~2)(h_1~2(7m~2 - 12 mn + 7n~2) + h_2~2(3m~2 - 4mn + 3n~2) - 2h_3~2(m~2 - 4mn + n~2)),Especially,for the smallest eigenvalue lambda_(min)pi~2/S~2,(11h_1~2+7h_2~2+6h_3~2)/24, where S is the area of the triangle with three lengths h_1≤h_2≤h_3.And it can be as a new quality of 2-D triangle grid for 2-nd PDE problems as q(tau):=(3h_3~2)/(16S~2)(11h_1~2+7h_2~2+6h_3~3)/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.
Language: 中文
Content Type: 期刊论文
URI: http://ir.iscas.ac.cn/handle/311060/14630
Appears in Collections:并行计算实验室 _期刊论文

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Recommended Citation:
孙家昶. 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]. Mathematica Numerica Sinica,2012-01-01,34(1):40932.
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