椭圆特征值问题基于高斯点的一种有效的谱配置法  

An effective spectral collocation method based on Gauss points for ellipse eigenvalue problems

作　　者：王国琳 安静 WANG Guolin;AN Jing(School of Mathematical Science,Guizhou Normal University,Guiyang,Guizhou 550025,China)

机构地区：[1]贵州师范大学数学科学学院

出　　处：《贵州师范大学学报（自然科学版）》2019年第6期73-77,共5页Journal of Guizhou Normal University：Natural Sciences

基　　金：贵州省科技计划项目(黔科合平台人才[2018]5769-01)

摘　　要：椭圆特征值问题基于高斯点的一种有效的谱配置法被提出。该方法首先利用Legendre多项式的性质构造一组满足边界条件的基函数,将逼近解由这组基函数展开。其次,利用正交多项式的三项递推关系,编程求解出每个基函数在这些高斯点处的节点值,将离散格式转化为一个线性特征系统。然后利用预条件迭代方法可快速地计算出逼近特征值和相应的特征向量。最后,分别对一维四阶椭圆特征值问题和二维二阶椭圆特征值问题给出了数值试验,数值结果表明该方法是非常有效的。An effective spectral collocation method based on Gauss points is proposed for ellipse eigenvalue problems.Firstly,according to the orthogonal properties of Legendre polynomials,we construct a set of basis functions which satisfy the boundary conditions.Then we expand the approximation solution by the set of basis functions.Secondly,using the three-item recursive relationship of the orthogonal polynomials,we compute the node values at each Gauss point for all the base functions and transform the discrete scheme into a linear eigen-system.Thirdly,by using the preconditioned iterative method,we can quickly compute the approximation eigenvalues and the corresponding eigenvectors.Finally,numerical experiments are given for the one-dimensional fourth-order ellipse eigenvalue problems and the two-dimensional second-order ellipse eigenvalue problems.The numerical results show that the method is very effective.

关 键 词：椭圆特征值问题 谱高斯配点法 预条件迭代法 数值实验 

分 类 号：O241.82[理学—计算数学] O1.[理学—数学]