Laplace Green's functions for infinite ground planes with local roughness

The Green's functions for the Laplace equation satisfying Dirichlet and Neumann boundary conditions on upper side of infinite plane with a circular hole are introduced studied. These enable solutions value problems in domains where is closed by an arbitrary mesh (locally rough surfaces). developed approach accounts positive negative ground elevations inside domain interest, which not possible to achieve using regular method images. Such appear electrostatics, however, methods apply other or Poisson equations govern. Integral series representations provided. Using these functions, efficient computational technique based element fast multipole acceleration developed. A numerical study some benchmark presented.