  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.
