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

Elliptic interface boundary value problems play a major role in numerous applications involving heat, fluids, materials, and proteins, to name a few. As an example, in implicit variational solvation, for the construction of biomolecular shapes, the electrostatic contributions satisfy the Poisson-Boltzmann equation with discontinuous dielectric constants across the interface. When interface motions are involved, one often needs not only accurate solution values, but accurate derivatives as well, such as the normal derivatives at the interface. We introduce here the Compact Coupling Interface Method (CCIM), a finite difference method for the elliptic interface problem with interfacial jump conditions. The CCIM can calculate solution values and their derivatives up to second-order accuracy in arbitrary ambient space dimensions. It combines elements of Chern and Shu's Coupling Interface Method and Mayo's approach for elliptic interface boundary value problems, leading to more compact finite difference stencils that are applicable to more general situations. Numerical results on a variety of geometric interfacial shapes and on complex protein molecules in three dimensions support the efficacy of our approach and reveal advantages in accuracy and robustness.