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 λm,n≈π2/24S2(h12(7m2-12mn+7n2)+h22(3m2-4mn+3n2)-2h32(m2-4mn+n2)),Especially, for the smallest eigenvalue λmin≈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.
. 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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