任意三角形Laplace特征值问题谱方法的数值对比研究

NUMERICAL COMPARISON RESEARCH OF LAPLACE EIGENVALUE ON ARBITRARY TRIANGLE USING SPECTRAL METHOD

本文选取多项式、有理多项式以及三角函数等五类函数作为基函数，设计相应的谱方法逼近格式并实现相应算法，对任意三角形上Laplace特征值问题进行数值求解对比研究．比较实验结果显示，谱方法相较于经典有限差分、有限元等低阶方法有较多的可信特征值；其中的Koornwinder多项式谱方法与基于Koornwinder多项式的有理谱方法，其可信特征值的数量达到全部计算特征值的4/π^2，并且达到“指数阶收敛”；而三角函数谱方法，则保持了稳定的收敛阶且有较多的可信特征值．

By choosing five different trial functions, including polynomials, rational polynomials and trigonometric functions, and designing the approximation scheme of spectral method, we compared the numerical results for Laplace eigenvalue problem on arbitrary triangle in this paper. The comparative numerical results show that the spectral method has more reliable eigenvalues compared with low-order methods such as classical finite difference method and finite element method. For the spectral methods based on Koornwinder polynomials and rational polynomials, the number of reliable eigenvalues can reach 4/π^2 of all calculating eigenvalues and the two methods can achieve exponential convergence. While the spectral method based on trigonometric functions keeps the stability of the order of convergence and has more reliable eigenvalues.