分片常系数电导率问题的边界元配点法

Collocation boundary element method for the conductivity equation with piecewise constant coefficient

用边界元方法讨论了具有分片常系数电导率方程Δ(γΔu)=0的Dirichlet边值问题.由于方程的基本解无法显式写出,在应用通常边界元时存在很大的困难.基于这个电导率方程的解的积分表达式,导出一个在边界和交界面上的积分方程组,并讨论了这个方程组的性质,对于这个积分方程组,用配点法进行求解,且给出其误差分析.相应的数值例子证实了算法的有效性.应该指出的是本文所用的方法也适用于具有分片常系数椭圆方程的不同边界问题.

The paper proposes a collocation BEM method for the Dirichlet problem of the conductivity equation Δ( γ Δ u )=0 which has the piecewise constant coefficient.Since the fundamental solution of this conductivity equation can not be expressed explicitly,it is difficult to apply the usual BEM method for this problem. Based on the integral representation formula for solutions of the conductivity equation, the integral equations on the boundary and interface curves are obtained and the properties of these integral equations are discussed. For the integral equations, a collocation method is proposed and the error analysis is given.The numerical simulation results show that this method is effective and accurate.It should be remarked that this method can be extended to treat other boundary value problems for elliptic equations with piecewise constant coefficient.