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Elliptic interface boundary value problems play a major role in numerousapplications involving heat, fluids, materials, and proteins, to name a few. Asan example, in implicit variational solvation, for the construction ofbiomolecular shapes, the electrostatic contributions satisfy thePoisson-Boltzmann equation with discontinuous dielectric constants across theinterface. When interface motions are involved, one often needs not onlyaccurate solution values, but accurate derivatives as well, such as the normalderivatives at the interface. We introduce here the Compact Coupling InterfaceMethod (CCIM), a finite difference method for the elliptic interface problemwith interfacial jump conditions. The CCIM can calculate solution values andtheir derivatives up to second-order accuracy in arbitrary ambient spacedimensions. It combines elements of Chern and Shu's Coupling Interface Methodand Mayo's approach for elliptic interface boundary value problems, leading tomore compact finite difference stencils that are applicable to more generalsituations. Numerical results on a variety of geometric interfacial shapes andon complex protein molecules in three dimensions support the efficacy of ourapproach and reveal advantages in accuracy and robustness.

A Compact Coupling Interface Method with Accurate Gradient Approximation for Elliptic Interface Problems