一个扩散问题的自然边界元法与有限元法组合

本文讨论由Helmholtz方程描述的扩散问题的自然边界元法与有限元法的组合．取一个圆作为公共边界，用Fourier展开建立边界积分方程，将无界区域上的问题化为有界区域上的非局部边值问题．在变分方程中公共边界上的未知量只包含函数本身而不包含其法向导数，从而减少了未知数的数目，并且边界元剐度矩阵只有极少量不同的元素，有利于数值计算．这种组台方法优越于建立在直接边界元法基础上的组合方法．文中证明了变分解的唯一性，数值解的收敛性和误差估计．最后讨论了数值技术并给出一个算倒．

THE COUPLING OF NATURAL BEM AND FEM FOR A SCATTERING PROBLEM

The coupling of natural boundary element method(BEM) and finite element method(FEM) for the scattering problem expressed by Helmholtz equation is discussed. A circle is taken as the common boundary, the integral equation is set up by Fourier expansion and the problem in an unbounded domain is transformed into the nonlocal boundary value problem in a bounded domain. Only the function itself, not its normal derivative at the common boundary, appears in the variational equation, so that the unknown numbers are reduced and the boundary element matrix has few different elements. Such a coupled method is superior to the one based upon the direct boundary element method. This paper proves the uniqueness for the variational solution, convergence and the error estimate for the numerical solution, discusses the numerical technique, and gives a numerical example.