Laplace Green's Functions for Infinite Ground Planes with Local Roughness

The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables solution of the boundary value problems in domains where the hole is closed by any surface. This approach enables accounting for arbitrary positive and negative ground elevations inside the domain of interest, which, generally, is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply to other domains where the Laplace or Poisson equations govern. Integral and series representations of the Green's functions are provided. An efficient computational technique based on the boundary element method with fast multipole acceleration is developed. A numerical study of some benchmark problems is presented.