  We propose a mesh refinement technique for solving elliptic difference
equations on unbounded domains based on the fast lattice Green's function
(FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh
and uses the fast multipole method in conjunction with fast Fourier transforms
to yield linear complexity and decrease time-to-solution. We extend this method
to a multi-resolution scheme and allow for locally refined Cartesian blocks
embedded in the computational domain. Appropriately chosen interpolation and
regularization operators retain consistency between the discrete Laplace
operator and its inverse on the unbounded domain. Second-order accuracy and
linear complexity are maintained, while significantly reducing the number of
degrees of freedom and hence the computational cost.
