任意凸四边形区域上二阶椭圆特征值问题基于高阶多项式逼近的一种数值方法
A Numerical Method Based on Higher Order Polynomial Approximation for Second Order Elliptic Eigenvalue Problems on Arbitrary Convex Quadrilateral Domain
摘要: 提出了任意凸四边形区域上二阶椭圆特征值问题基于高阶多项式逼近的一种有效的数值方法。首先，利用等参变换将任意凸四边形区域上的函数转化为变[-1,1]Χ[-1,1]上的函数，并建立原问题在等参变换下的弱形式及其逼近格式。其次，利用Legendre正交多项式的性质构造逼近空间中有效的一组基函数，将逼近格式转化为基于矩阵形式的线性特征系统，从而可以通过MATLAB软件编程求解出相应的特征值。最后，一些数值算例被呈现，数值结果进一步验证了我们算法的有效性和收敛性。
Abstract: An effective numerical method based on high-order polynomial approximation for second-order elliptic eigenvalue problems on arbitrary convex quadrilateral regions is proposed. Firstly, the function on any convex quadrilateral region is transformed into a function on variable by isoparametric transformation, and the weak form and approximation scheme of the original problem under isoparametric transformation are established. Secondly, a set of effective basis functions in the approximation space are constructed by using the properties of Legendre orthogonal polynomials, and the approximation format is transformed into a linear characteristic system based on matrix form, so that the corresponding eigenvalues can be solved by MATLAB software programming. Finally, some numerical examples are presented, and the numerical results further verify the effectiveness and convergence of our algorithm.
文章引用：郑继会. 任意凸四边形区域上二阶椭圆特征值问题基于高阶多项式逼近的一种数值方法[J]. 应用数学进展, 2021, 10(12): 4201-4208. https://doi.org/10.12677/AAM.2021.1012446

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