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
查看参考文献20篇
|
文摘
|
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≈π2/24S2（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λ_(min)=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（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. |
|
其他语种文摘
|
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）≈π~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 λ_（min）≈π~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（τ）:=（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. |
|
来源
|
计算数学
,2012,34(1):1-24 【核心库】
|
|
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
|
|
地址
|
中国科学院软件研究所并行计算实验室, 北京, 100190
|
|
语种
|
中文 |
|
文献类型
|
研究性论文 |
|
ISSN
|
0254-7791 |
|
学科
|
数学 |
|
基金
|
国家自然科学基金
|
|
文献收藏号
|
CSCD:4440264
|
参考文献 共
20
共1页
|
|
1.
Li H. Discrete Fourier analysis, cubature and interpolation on a hexagon and a triangle.
SIAM J. Numer. Anal.,2008,46:1653-1681
|
被引
7
次
|
|
|
|
|
2.
Li Peter. On the Schrodinger equation and the Eigenvalue problem.
Comm. Math. Phys.,1983,88:309-318
|
被引
1
次
|
|
|
|
|
3.
Pinsky Mark A. The eigenvalues of an equilateral triangle.
SIAM J. Math. Anal.,1980,11:819-827
|
被引
3
次
|
|
|
|
|
4.
Pinsky Mark A. Completeness of the eigenfunctions of the equilateral triangle.
SIAM J. Math. Anal.,1985,16:848-851
|
被引
4
次
|
|
|
|
|
5.
Prager Milan. An investigation of eigenfunctions over the equilateral triangle and square.
Applications of mathematics,1998,43:311-320
|
被引
5
次
|
|
|
|
|
6.
McCartin B J. Eigenstructure of the equilateral triangle. I. The Dirichlet problem.
SIAM Rev.,2003,45:267-287
|
被引
4
次
|
|
|
|
|
7.
Sun Jiachang. ORTHOGONAL PIECE-WISE POLYNOMIALS BASIS ON AN ARBITRARY TRIANGULAR DOMAIN AND ITS APPLICATIONS.
Journal of Computational Mathematics,2001,19:55-66
|
被引
8
次
|
|
|
|
|
8.
Sun J. Multivariate Fourier series over a class of non tensor-product partition domains.
Journal of Computational Mathematics,2003,21:53-62
|
被引
9
次
|
|
|
|
|
9.
Sun Jiachang. Generalized Fourier transform on an arbitrary tirangular domain.
Advances in Computational Mathematics,2005,2:223-248
|
被引
5
次
|
|
|
|
|
10.
Laugesen R S.
Dirichlet eigenvalue sums on triangles are minimal for equilaterals. preprint,2011
|
被引
1
次
|
|
|
|
|
11.
Laugesen R S. Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals.
Mathematical Inequalities & Applications, preprint,2011
|
被引
1
次
|
|
|
|
|
12.
Laugesen R S. Sums of Laplace eigenvalues-rotationally symmetric maximizers in the plane.
J. Funct. Anal.,2011,260:1795-1823
|
被引
1
次
|
|
|
|
|
13.
Laugesen R S. Maximizing Neumann fundamental tones of triangles.
J. Math. Phys.,2009,50:112903
|
被引
1
次
|
|
|
|
|
14.
Weideman J A C. The eigenvalues of second-order spectral differentiation matrices.
SIAM J. Numer. ANAL.,1988,25(6):1279-1298
|
被引
4
次
|
|
|
|
|
15.
孙家昶.
非规则区域Fourier变换与正交多项式,2009
|
被引
5
次
|
|
|
|
|
16.
Sun Jiachang.
PDE Eigen-polynomials over curved triangle domains and their recurrence relations RDCPS-AR-09-02, RDCPS Annual Reports, 20,2009:15-31
|
被引
1
次
|
|
|
|
|
17.
Sun Jiachang.
On PDE eigen-polynomial over a bivariate 5-degree domain and their recurrences relations. RDCPS-AR-09-01, RDCPS Annual Reports, 20,2009:1-14
|
被引
2
次
|
|
|
|
|
18.
Sun Jiachang.
On PDE eigen-polynomial over a bivariate 5-degree domain and their recurrences relations. RDCPS-AR-09-01, RDCPS Annual Reports, 20,2009:1-14
|
被引
2
次
|
|
|
|
|
19.
杨志杰. 任意三角形区域中一组完备正交基的构造与分类.
计算数学,2003,25(2):219-230
|
被引
2
次
|
|
|
|
|
20.
孙家昶. 特征值问题的预变换方法(I):杨辉三角阵变换与二阶PDE特征多项式.
中国科学：数学,2011,41(8):701-724
|
被引
5
次
|
|
|
|
|
了解气象卫星更多信息，请访问