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

TL;DR

This work tackles elliptic interface problems with jump conditions, where accurate gradients are essential for interface dynamics. It introduces the Compact Coupling Interface Method (CCIM), a finite difference framework that merges CIM with Smereka's second-order discrete delta approach to deliver second-order accuracy for both the solution and its gradient on complex, potentially moving interfaces. CCIM achieves a more compact stencil, handles mixed derivatives with multiple schemes, and leverages first-order derivative information to eliminate exceptional points, providing robust convergence on 3D biomolecular surfaces and high-contrast scenarios. The method is demonstrated on geometric and protein surfaces, including moving interfaces, with publicly available code and clear implications for electrostatics and interface-fluid-structure problems.

Abstract

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.

Figures (16)

• Figure 1: Schematic for the elliptic interface problem. $\Gamma$ is an interface that separates a cubical domain $\Omega$ with boundary $\partial \Omega$ into $\Omega^+$ and $\Omega^-$. The normal to the interface is denoted as $\vb n$. The dashed lines are the grid lines of the uniform mesh. The circles are interiors points, where standard central difference can be used. The disks are examples of on-front points, where special stencils are required.
• Figure 2: Examples of a (1) CIM2 stencil and a (2) CCIM stencil at $\vb{x_i}$. The circles and disks are grid points on different sides of the interface. The asterisks are the intersections of the surface and the grid lines. In this case, CIM2 and ICIM have the same stencil, which requires 2 points on the same side of the interface in each dimension chernCouplingInterfaceMethod2007. Points that does not satisfy this requirement are handled in ICIM shuAccurateGradientApproximation2014. The CCIM stencil requires fewer points so it's more compact. Both CIM2 and CCIM need some extra grid points compared to the standard central difference stencil.
• Figure 3: The interface intersects the grid segment $\overline{ \vb{x_i} \vb x_{\vb i + \vb e_k}}$ at ${\vb{\hat{x}}_k}$. $u^-$ and $u^+$ are the limits of u at ${\vb{\hat{x}}_k}$ from $\Omega^-$ and $\Omega^+$. $u_{\vb i}$ and $u_{\vb{i}+\vb{e}_k}$ are approximated by Taylor's expansion at the interface.
• Figure 4: Approximation of the mixed derivative $\pdv*[2]{u}{x_k}{x_l}$ at ${\vb{x_i}}$. The circles and disks are grid points in $\Omega^-$ and $\Omega^+$. The $u$-values at the squares are used to approximate the mixed derivative. Case 1 is the usual central difference. Case 2 and 3 are biased difference. Case 4 uses the first-order derivatives at at ${\vb{x_i}}$. Case 5 uses the first-order and the second-order derivatives at at ${\vb{x_i}}$. Case 6 uses the jump $[\pdv*[2]{u}{x_k}{x_l}]$ at ${\vb{\hat{x}}_k}$ and the mixed second-order derivative on the other side at $\vb x_{\vb i + \vb e_k}$, which is approximated by central difference.
• Figure 5: An example of CCIM stencil at $\vb{x_i}$. The grid points in the stencil are labelled from 1 to 12. The disks and the circles are the grid points outside and inside the surface. (1,10,11) are used to approximate $u_{yz}(\vb{x_i})$ (see Fig. \ref{['f:mixdu']} case 4). (1, 3, 12) are used to approximate $u_{zx}(\vb{x_i})$ (Fig. \ref{['f:mixdu']} case 5). (8, 9, 10, 11) can be used to approximate $u_{xy}(\vb{x_i})$. Alternatively, $u_{xy}(\vb{x_i})$ can be approximated by using $u_{xy}$ at the other side (any one of 2, 4, 5, 6 or 7) and the jump condition (see Fig. \ref{['f:mixdu']} case 6).