边界积分方程解的一个后验误差估计

从边界元法导出的边界积分方积的精确解通常是求不出的,于是其近似解的实际误差是无法得到的。本文说明在H^1/2范数里,近似解的剩余误差可以用作误差估计,以一条弧为边界的Helmholtz方程外Dirichlet问题导出的边界积分方程为例,分别用一般的边界元法和带奇性单元的边界元法进行计算。数值结果显示奇性单元的应用使误差明显减小,并且乘余误差的H^0范数十分接近H^1/2范数。

A Posterior Estimate of Error in Solution of Some Boundary Integral Equations

Since the exact solution of a boundary integral equation from BEM is unknown, the actual error in the approximate solution cannot be obtained. In this paper, it is shown that the residual error in the H 1/2 norm can be used as an estimate of error. The test problem is the boundary integral equation from the exterior Dirichlet problem for the Helmholtz equation with a smooth arc as its boundary, and it is solved using BEM both with and without the singular elements. The numerical results show that the use of the singular elements reduces the error significantly, and that the residual error in the H 0 norm approximates to one in the H 1/2 norm.