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  

PRE-TRANSFORMED METHODS FOR EIGEN-PROBLEMSⅡ:EIGEN-STRUCTURE FOR LAPLACE EIGEN-PROBLEM OVER ARBITRARY TRIANGLES

作　　者：孙家昶[1] 

机构地区：[1]中国科学院软件研究所并行计算实验室,北京100190

出　　处：《计算数学》2012年第1期1-24,共24页Mathematica Numerica Sinica

基　　金：国家自然基金项目资助(No.60970089,61170075)

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≈π~2/24S^2(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λmin=λ_(2,1)≈π~2/S^2 11h_1~2+7h_2~2+6h_3~2/24,其中S是该三角形(h1≤h2≤h3)的面积,可作为数值PDE中三角剖分质量的一种新标准q(ζ):=3h_3~2/16S^2 11h_3~2+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.Based on Laplace eigen-structure over three special triangle domains （regular trian- gle, isoceles triangle and triangle with （30°, 60°, 90°））, we propose a unified basis to com- pute 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 asλm,n≈π2/24s2（h1^2（7m^2-12mn＋7n^2）=h2^2（3m^2-4mn＋3n^2）-2h2^2（m^2-4mn＋n^2））,Especially, for the smallest eigenvalueλmin=λ2,1≈χ2/s211h1^2＋7H2^2＋6H3^2/24 ~ ＂ lengths where S is the areu of the triangle with three h1≤h2≤h3.And it can be as a new quality of 2-D triangle grid for 2-nd PDE problems as q（T）：=3h3^2/16s^211h1^1＋7h2^2＋6h3^2/24/Toreflect 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.

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 

分 类 号：O177[理学—数学]