Galerkin Boundary Element Method With Hyper Singular Intergral

The Neumann Problem of Laplace equation is an important elliptic boundary value problem with extensive applications。When the direct boundary integral formula in Calderon projection applied to solve this problem, we shall encounter the integral with hypersingularity. Many approaches have been proposed from different points of view to overcome the difficulties of computing hyper singular integrals. We apply a Galerkin Boundary Elements to solve the integral equation of this kind.The scheme is being known as the regularization of divergent integral in distribution, We use the concept of integration by parts in the sense of distributions, when boundary rotation is introduced, the two order normal derivatives of singular kernel are shifted to the boundary rotation of unknown function in the Galerkin variational formulation.Nedelec introduced this scheme and deduced complete numerical computation formula with boundary rotation and a weak singular integral kernel for 3 dimensional case. As for the 2-D problem, Zhu Jialin proposed a variational formulation, but without implementation, ZHANG Shou-gui completed the numerical implementation. The approach can be used to overcome the difficulty of integrals with hypersingularity on any arbitrary shaped smooth boundary.Another approaches to overcome the hyper singularity, which are based on the series development of integral kernel and the separation of the singular parts, or integration by parts with tangential derivatives presented by Yu Dehao, Han Houdehave been applied to the case of circular or rectangular boundary.While linear boundary elements are used for 2-dimensional problem, the boundary rotation on each element can be discretized into a constant vector, so that the integration can be performed in a simple way and the barrier of numerical calculation for integrals with hyper singularity is overcome. We write a program in Fortran90, and did some numerical tests, The numerical results of examples demonstrate that the scheme presented is practical and effective.