Helmholtz方程周期Green函数及其偏导数截断误差收敛阶的分析

ANALYSIS OF THE CONVERGENCE RATES FOR THE TRUNCATION ERRORS OF PERIODIC GREEN'S FUNCTION OF HELMHOLTZ EQUATION AND ITS PARTIAL DERIVATIVES

在应用边界元方法求解Helmholtz方程周期边值问题时,需要构造以周期Green函数或其偏导数为核函数的积分算子形式的解.由于Helmholtz方程的周期Green函数G^P是一个函数项级数,该级数的通项是Hankel函数,在数值求解中,需要对其进行截断,从而很有必要研究其截断误差.本文根据Hankel函数在变量趋于无穷大时的渐近展开式,并结合Abel不等式,证明了G^P及其一阶偏导和二阶混合偏导一致收敛,且其截断误差收敛阶均为O(1/p^(1/2)).最后,通过数值实验验证了理论证明的正确性.本文的证明方法也可被用于证明其它一些方程周期Green函数的收敛性问题.

To solve the numerical solutions of periodic boundary value problems of Helmholtz equations with periodic BEM, the periodic single and/or double layered potential will be constructed, the kernels of the periodic potentials are the periodic Green＇s function or its partial derivatives. The periodic Green＇s function GP of Helmholtz equation is the infinite summation of Hankel functions. In this paper, based on the asymptotic expansions of Hankel functions for large arguments, the mathematical proofs of the uniform convergence and convergence rates O（1/√P） of G^p and its derivatives are given by using the Abel＇s inequality. Finally, the conclusions are verified by some numerical tests. The proofs in this paper can also be used to prove the convergence of other periodic Green＇s functions.