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.

翻译：椭圆界面边值问题在热学、流体、材料及蛋白质等众多应用中扮演着重要角色。以隐式变分溶剂化模型为例，在构建生物分子形状时，静电贡献需满足界面处介电常数不连续的泊松-玻尔兹曼方程。当涉及界面运动时，往往不仅需要精确的解值，还需要精确的导数（如界面处的法向导数）。本文提出一种紧凑耦合界面方法（CCIM），这是一种用于具有界面跳跃条件的椭圆界面问题的有限差分方法。CCIM可在任意空间维度下计算解值及其一阶与二阶导数，精度达到二阶。该方法融合了Chern和Shu的耦合界面方法及Mayo处理椭圆界面边值问题的思路，通过更紧凑的有限差分模板实现了更广泛的应用场景。针对多种几何界面形状及三维复杂蛋白质分子的数值实验结果，证实了该方法的有效性，并揭示了其在精度和鲁棒性方面的优势。