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.

翻译：我们提出了一种紧致耦合界面方法（CCIM），这是一种有限差分方法，能够针对具有界面跳跃条件的椭圆复杂界面问题，获得不仅解值而且其梯度的二阶精确逼近。这类具有界面跳跃条件的椭圆界面边值问题在热传导、流体流动、材料科学和蛋白质对接等众多领域的关键应用中扮演着核心角色。一个典型的例子涉及生物分子形状的构建，其中此类椭圆界面问题以线性化泊松-玻尔兹曼方程的形式出现，该方程涉及跨越界面的不连续介电常数，并主导着静电贡献。此外，当涉及界面动力学时，界面的法向速度可能由解的法向导数构成，而我们的方法可以对此进行二阶逼近，从而获得精确的界面动力学。我们的方法可以在任意空间维度中构建，它结合了备受推崇的耦合界面方法（针对此类椭圆界面问题）和Smereka的二阶精确离散δ函数。其结果是形成了一种变体与混合方法，其模板比耦合界面方法中的更为紧致，并且在涉及几何模型问题和复杂生物分子表面的数值实验中，展现出更稳健的误差分布优势。