Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains
Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains
复制标题
盘域修正亥姆霍兹方程诺依曼问题的基本解法
DOI：
10.1016/j.cam.2021.113795
复制
发表时间：
2021
影响因子：
2.4
通讯作者：
Yoshitaro Tanaka,
中科院分区：
文献类型：
--
作者：
Shin-Ichiro Ei;Hiroyuki Ochiai;Yoshitaro Tanaka,
用基本解的方法构造有界域上偏微分方程的近似解。结合移到域外点的基本解，确定线性和的系数，在边界的有限点上满足边界条件。本文严格地证明了圆盘域上修正Helmholtz方程的Neumann问题的MFS近似解的存在性。给出了近似解存在的充分条件。将格林公式应用于修正Helmholtz方程的Neumann问题，通过边界积分将近似解与精确解之间的误差限定为边界条件函数与近似解的法向导数之差。利用这种误差估计，我们证明了MFS近似解收敛于指数阶的精确解，即N 2 a N阶，其中a是小于1的正常数，N是并置点的个数。此外，数值模拟结果表明，随着配点数N的增加，误差呈指数级趋近于0。
The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, N 2 a N order, where a is a positive constant less than one and N is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points N.
使用实部对偶边界元法对圆形腔的真实和虚假本征解进行分析研究和数值实验
DOI：
10.1002/1097-0207(20000730)48:9
发表时间：
2000
期刊：
影响因子：
--
作者：
S. Kuo;Jeng;C. X. Huang
通讯作者：
C. X. Huang