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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 | |
| Alternative Title | PRE-TRANSFORMED METHODS FOR EIGEN-PROBLEMSⅡ:EIGEN-STRUCTURE FOR LAPLACE EIGEN-PROBLEM OVER ARBITRARY TRIANGLES |
| 孙家昶 | |
| 2012 | |
| Source | Mathematica Numerica Sinica
 |
| ISSN | 0254-7791 |
| Volume | 34Issue:1Pages:40932 |
| English 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. |
| 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. |
| Keyword | Pre-transformed Eigenvalues Eigen-vectors Laplace Pde Eigen-problem Arbitrary Triangle |
| 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. |
| Subject | Mathematics |
| Language | 中文 |
| Content Type | 期刊论文 |
| URI | http://ir.iscas.ac.cn/handle/311060/14630 |
| Collection | 并行软件与计算科学实验室 |
| Recommended Citation GB/T 7714 | 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,2012,34(1):40932. |
| APA | 孙家昶.(2012).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.Mathematica Numerica Sinica,34(1),40932. |
| MLA | 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".Mathematica Numerica Sinica 34.1(2012):40932. |
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| 特征值问题的预变换方法(Ⅱ)任意三角形域（571KB） | 开放获取 | License | Application Full Text | |||
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