The numerical solution of exterior neumann problem for Helmholtzʹs equation via modified greenʹs functions approach

In the 1970s, modified Greenʹs function approach for solving the Helmholtz equation was proposed by Jones and Ursell and in the 1980s was clarified by Kleinman, Roach and Kress. To this date there are no numerical results available for this approach. In this paper, a global Galerkin method is used to numerically solve the exterior Neumann problem for the Helmholtz equation in three dimensions based on Jonesʹ modified integral equation approach. Jones approach directly leads to an integral equation which only involves weakly singular operators, thus is a good alternative for solving the exterior Neumann problem. Theoretical and computational details of the method are presented.