Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

Haibing Wang 1*, Jijun Liu 2

1 School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
2 Department of Mathematics, Southeast University, Nanjing 210096, China

Received February 20, 2006; Accepted (in revised version) October 29, 2006
Abstract

We consider the numerical solution for the Helmholtz equation in $\Bbb{R}^2$ with mixed boundary conditions. The solvability of this mixed boundary value problem is established by the boundary integral equation method. Based on the Green formula, we express the solution in terms of the boundary data. The key to the numerical realization of this method is the computation of weakly singular integrals. Numerical performances show the validity and feasibility of our method. The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.

Key words: Helmholtz equation; Green formula; potential theory; boundary integral equation; numerics.

AMS subject classifications: 35J05, 31A10, 65N99

Correspondence to: Haibing Wang , School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China Email: wanghb845@yahoo.com.cn