  In this paper we present a method to treat interface jump conditions for
constant coefficients Poisson problems that allows the use of standard "black
box" solvers, without compromising accuracy. The basic idea of the new approach
is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections
applied on nodes located across the interface for discretization stencils that
straddle the interface. If the corrections are solution-independent, they can
be moved to the right-hand-side (RHS) of the equations, producing a problem
with the same linear system as if there were no jumps, only with a different
RHS. However, achieving high accuracy is very hard (if not impossible) with the
"standard" approaches used to compute the GFM correction terms.
  In this paper we generalize the GFM correction terms to a correction
function, defined on a band around the interface. This function is then shown
to be characterized as the solution to a PDE, with appropriate boundary
conditions. This PDE can, in principle, be solved to any desired order of
accuracy. As an example, we apply this new method to devise a 4th order
accurate scheme for the constant coefficients Poisson equation with
discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil
discretization of the Poisson equation, (ii) a representation of the correction
function in terms of bicubics, and (iii) a solution of the correction function
PDE by a least squares minimization. Several applications of the method are
presented to illustrate its robustness dealing with a variety of interface
geometries, its capability to capture sharp discontinuities, and its high
convergence rate.
