基于多项式插值的有限差分法求解Helmholtz方程透射特征值问题
Solving the Transmission Eigenvalue Problem of Helmholtz Equation by Finite Difference Method Based on Polynomial Interpolation
摘要: 有限差分公式在无网格方法求解微分方程数值解中起着重要作用。本文针对Helmholtz方程透射特征值问题，通过多项式插值来创建有限差分公式。本文运用一种简单实用的节点分布，既保证多元多项式插值的唯一可解性，又使矩阵为三角矩阵，以便构造的基本多项式化为Lagrange基多项式。最后给出了外透射特征值问题的数值算例。
Abstract: Finite difference formulas play an important role in the numerical solution of differential equations using meshless methods. In this paper, aiming at the transmission eigenvalue problem of Helmholtz equation, a finite difference formula is created by polynomial interpolation. Then, a simple and practical node distribution is used, which not only guarantees the unique solvability of multivariate polynomial interpolation, but also makes the matrix a triangular matrix so that the basic polynomials constructed can be transformed into Lagrange basis polynomials. Finally, a numerical example solving the eigenvalue problem of external transmission is given.
文章引用：李悠然, 潘文峰. 基于多项式插值的有限差分法求解Helmholtz方程透射特征值问题[J]. 应用数学进展, 2020, 9(12): 2236-2243. https://doi.org/10.12677/AAM.2020.912261

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