  We propose the Compact Coupling Interface Method (CCIM), a finite difference
method capable of obtaining second-order accurate approximations of not only
solution values but their gradients, for elliptic complex interface problems
with interfacial jump conditions. Such elliptic interface boundary value
problems with interfacial jump conditions are a critical part of numerous
applications in fields such as heat conduction, fluid flow, materials science,
and protein docking, to name a few. A typical example involves the construction
of biomolecular shapes, where such elliptic interface problems are in the form
of linearized Poisson-Boltzmann equations, involving discontinuous dielectric
constants across the interface, that govern electrostatic contributions.
Additionally, when interface dynamics are involved, the normal velocity of the
interface might be comprised of the normal derivatives of solution, which can
be approximated to second-order by our method, resulting in accurate interface
dynamics. Our method, which can be formulated in arbitrary spatial dimensions,
combines elements of the highly-regarded Coupling Interface Method, for such
elliptic interface problems, and Smereka's second-order accurate discrete delta
function. The result is a variation and hybrid with a more compact stencil than
that found in the Coupling Interface Method, and with advantages, borne out in
numerical experiments involving both geometric model problems and complex
biomolecular surfaces, in more robust error profiles.
