中文摘要：

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≈π2/24s2（h1^2（7m^2-12mn＋7n^2）=h2^2（3m^2-4mn＋3n^2）-2h2^2（m^2-4mn＋n^2））,（m〉n≥）特别，对于最小特征值λmin=λ2,1≈χ2/s211h1^2＋7H2^2＋6H3^2/24,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咱新标准q（T）：=3h3^2/16s^211h1^1＋7h2^2＋6h3^2/24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．

英文摘要：

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.