Laplace方程的Galerkin边界元解法

Galerkin Boundary Element Method for Laplace Equation

Galerkin方法是基于变分原理基础上的一种把微分方程或积分方程转化为等价的变分方程,通过离散变分方程求原方程数值解的数值计算方法。把Laplace方程的边值问题转化为边界积分方程后,通过与边界积分方程等价的变分形式,采用线性单元,利用Galerkin边界元方法求解。在计算单元刚度矩阵时,对二重积分的第一重使用精确积分,第二重使用数值积分,从而有效克服了奇异积分的计算,数值算例验证了Galerkin方法误差的理论结果。

Galerkin method based on the variation principle is used to solve differential and integral equations. The boundary problem of Laplace equation is changed into the variational equation which is equivalent to the boundary integral equation. Using linear element, it is solved by Galerkin boundary element method. In computation of stiffness matrix, the exactly integral formula is used in the first order integral expression, The numerical integral formula is used in the second order integral expression. Thus the problem of calculation of double singular integral is carried out. The numerical experiments also prove this method is reliable. The error of Galerkin boundary element is tested with numerical experimentation.