三维Helmholtz方程Dirichlet问题的边界元法及其收敛性分析

The BEM for Dirichlet Problems of Three Dimensional Helmholtz Equation and Its Convergence

本文把三维Ｈｅｌｍｈｏｌｔｚ方程Ｄｉｒｉｃｈｌｅｔ内外边值问题结合起来讨论，得到统一形式的边界积分方程。首先，据拟微分算子的理论，讨论了积分算子的性质及问题弱解的存在唯一性；接着采用边界元方法，离散积分方程得到数值解；最后，给出了解的全局误差估计及内部超收敛估计。

In this paper, we discuss the combination of the interior and exterior boundary values of Dirichlet problem for three dimensional Helmholtz equation, and get the uniform boundary integral equations. First, according to the theory of pseudo-differential operator,we study the properties of integral operator and the existence of weak solution, secondly, by use of BEM, we discret the integral equation and obtain the numerical solution. Finally, the global error estimation and interior superconvergence estimation are given.