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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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