Discrete Green's function diakoptics — A toy problem

In Green's function diakoptics, wave-field interactions between disjoint domains in space are described in terms of interacting multi-port subsystems. In addition to integral-equation based Green's function diakoptics for time-harmonic fields, a demonstrably stable discrete space-time Green's function diakoptics scheme has been reported to be effective for proximate regions. For distant regions it is necessary to retain a long field history. Upon considering a diakoptics toy problem involving two disjoint domains consisting of a single point each, the resulting boundary operator may be expressed in terms of two distinct discrete Green's functions only, describing the intra- and inter-domain unit-source field responses, respectively. There are various integral representations and asymptotic expansions for the discrete Z-domain Green's functions, and it is even possible to construct representations for the discrete space-time Green's functions consisting of nested finite sums, which turn out to be difficult to evaluate due to severe cancellations.