Galerkin Boundary Element Method For 2-D Laplace Equation Of Neumann Problem

The Neumann Problem of Laplace equation is an important elliptic boundary value problem and has extensive applications. The solution will lead to solve a hyper singular integral equation, when a double layer potential distribution formulation is used. Many approaches have been proposed from different points of view to overcome the difficulties of computing hyper singular integrals In the present work, we propose a variational formulation in the sense of distributions. In this scheme, being known as the regularization of divergent integral in distribution, the partial derivatives of singular kernel are shifted to the unknown function in the variational formulation, thus the order of singularity is reduced by one by performing one integration by part. As the process of deduction is complicated, less people used it in the real situation of numerical computing. Nedelec introduced this scheme and formed complete numerical computation formula with boundary rotation and a weak singular integral kernel for 3 dimensional case. For 2-D problem, Zhu Jialin proposed a variational formulation, but without detailed computation formulation. As for the numerical implementation of this scheme applied to any arbitrary shaped boundary, we have not found any report on this issue. Another approach to overcome the hyper singularity, which is based on the series development of integral kernel and the separation of the singular parts, presented by Yu Dehao, Han Houde have been applied to the case of circular or rectangular boundary. In this paper, the Galerkin approach is used to solve the hyper singular integral equation associated with the double layer solution of the Neumann problem of Laplace equation, the variational formulation with hyper singular integral kernel is changed into a weak one with boundary rotation. When linear boundary elements are used, the boundary rotation can be discretized into a simple form, so that the integration can be performed in a simple way. The results of numerical examples demonstrate that the scheme presented is practical and effective.