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

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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:λ_m,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λ_min=λ_2,1≈π²/S² 11h₁²+7h₂²+6h₃²/24,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(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 λ_m,n≈π²/24S²(h₁²(7m²-12mn+7n²)+h₂²(3m²-4mn+3n²)-2h₃²(m²-4mn+n²)),Especially, for the smallest eigenvalue λ_min≈π²/S² 11h₁²+7h₂²+6h₃²/24,where S is the area of the triangle with three lengths h₁≤h₂≤h₃.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 words： Pre-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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[1]	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]. 计算数学, 0, (): 1-24.