  Lattice Green's Functions (LGFs) are fundamental solutions to discretized
linear operators, and as such they are a useful tool for solving discretized
elliptic PDEs on domains that are unbounded in one or more directions. The
majority of existing numerical solvers that make use of LGFs rely on a
second-order discretization and operate on domains with free-space boundary
conditions in all directions. Under these conditions, fast expansion methods
are available that enable precomputation of 2D or 3D LGFs in linear time,
avoiding the need for brute-force multi-dimensional quadrature of numerically
unstable integrals. Here we focus on higher-order discretizations of the
Laplace operator on domains with more general boundary conditions, by (1)
providing an algorithm for fast and accurate evaluation of the LGFs associated
with high-order dimension-split centered finite differences on unbounded
domains, and (2) deriving closed-form expressions for the LGFs associated with
both dimension-split and Mehrstellen discretizations on domains with one
unbounded dimension. Through numerical experiments we demonstrate that these
techniques provide LGF evaluations with near machine-precision accuracy, and
that the resulting LGFs allow for numerically consistent solutions to
high-order discretizations of the Poisson's equation on fully or partially
unbounded 3D domains.

Numerical Analysis;Numerical Analysis