A Compact Coupling Interface Method with Second-Order Gradient Approximation for Elliptic Interface Problems

Abstract

We propose the Compact Coupling Interface Method, 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.

Appendices

Differentiation of the Jump Condition

In this appendix, we detail the calculation of the formula for approximation of $[\nicefrac {\partial ^{2} u}{\partial x_k^{2}}]$, through equations involving terms of $[\nabla ^2 u]$, as found in Sect. 2.4. In the following derivation, quantities related to f, a, $\epsilon$ and $\tau$ are all known. Our final goal is to write the jump of the second derivatives $[\nabla ^2u]$ in terms of the known quantities and the one-sided derivatives $\nabla u^-$, $\nabla ^2u^-$. Since our interface is smooth, we can consider any smooth extension of $\tau$ and $\sigma$ off the interface, therefore quantities such as $\nabla ^2\tau$ and $\nabla \tau \cdot {\textbf{n}}$ are well-defined.

We first consider jumps of the first derivatives of u at the interface, especially in terms of jumps of the normal and tangential derivatives of u. Taking tangential derivatives on both sides of $u^+-u^- = [u] = \tau$, we get

$$\begin{aligned}{}[\nabla u\cdot {\textbf{s}}_j] = \nabla u^+\cdot {\textbf{s}}_j-\nabla u^-\cdot {\textbf{s}}_j = \nabla \tau \cdot {\textbf{s}}_j, \end{aligned}$$

for $j = 1,\dots ,d-1$. On the other hand, with

we can get, when $v = \nabla u\cdot n$,

These equations for the jumps of the normal and tangential derivatives of u can then be used to get $[\nabla u]$, from

Thus having handled jumps of first derivatives of u, we now turn our attention jumps of second derivatives and derive equations for the terms of $[\nabla ^2 u]$ in three ways.

Tangential derivative of jump of tangential derivative: We can get equations on terms of $[\nabla ^2 u]$ by starting with the jump of the tangential derivative of u along ${\textbf{s}}_m$, namely $[\nabla u\cdot s_m]$, and taking its tangential derivative along ${\textbf{s}}_n$, giving $\nabla [\nabla u\cdot s_m]\cdot s_n$. This quantity can be written as

$$\begin{aligned} \nabla \left[ \nabla u \cdot {\textbf{s}}_m\right] \cdot {\textbf{s}}_n = \nabla (\nabla \tau \cdot {\textbf{s}}_m) \cdot {\textbf{s}}_n. \end{aligned}$$

However, we also have, using (11), that

$$\begin{aligned} \nabla \left[ \nabla u \cdot {\textbf{s}}_m\right] \cdot {\textbf{s}}_n= & {} {\textbf{s}}_n^T \left[ \nabla ^2 u\right] {\textbf{s}}_m + {\textbf{s}}_n^T\nabla {\textbf{s}}_m \left[ \nabla u\right] \\= & {} {\textbf{s}}_n^T \left[ \nabla ^2 u\right] {\textbf{s}}_m + \frac{1}{\epsilon ^+} \left( \sigma - \left[ \epsilon \right] \nabla u^-\cdot {\textbf{n}}\right) {\textbf{s}}_n^T\nabla {\textbf{s}}_m {\textbf{n}} + \sum _{j=1}^{d-1} (\nabla \tau \cdot {\textbf{s}}_j) {\textbf{s}}_n^T \nabla {\textbf{s}}_m {\textbf{s}}_j \end{aligned}$$

and, additionally, that

$$\begin{aligned} \nabla (\nabla \tau \cdot {\textbf{s}}_m) \cdot {\textbf{s}}_n= & {} {\textbf{s}}_n^T \nabla ^2 \tau {\textbf{s}}_m + s_n^T \nabla {\textbf{s}}_m \nabla \tau \\= & {} {\textbf{s}}_n^T \nabla ^2 \tau {\textbf{s}}_m + (\nabla \tau \cdot {\textbf{n}}) {\textbf{s}}_n^T \nabla {\textbf{s}}_m {\textbf{n}} + \sum _{j=1}^{d-1} (\nabla \tau \cdot {\textbf{s}}_j) {\textbf{s}}_n^T \nabla {\textbf{s}}_m {\textbf{s}}_j. \end{aligned}$$

Thus, equating these and solving for the term with jumps in second derivatives of u, especially simplifying using the fact that

$$\begin{aligned} \nabla {\textbf{n}}~ {\textbf{s}}_j = - \nabla {\textbf{s}}_j {\textbf{n}}, \quad j = 1, \dots , d-1, \end{aligned}$$

from taking the gradient on both sides of ${\textbf{n}} \cdot {\textbf{s}}_j = 0$ for $j = 1, \dots , d-1$, we get (18):

$$\begin{aligned} {\textbf{s}}_n^T [\nabla ^2u]{\textbf{s}}_m = {\textbf{s}}_n^T \nabla ^2 \tau {\textbf{s}}_m - \frac{1}{\epsilon ^+}( \sigma - [\epsilon ] \nabla u^- \cdot {\textbf{n}}) {\textbf{s}}_n^T \nabla {\textbf{n}}{\textbf{s}}_m - (\nabla \tau \cdot {\textbf{n}}) {\textbf{s}}_n^T \nabla {\textbf{n}}{\textbf{s}}_m. \end{aligned}$$

Tangential derivative of flux jump: For more equations on terms of $[\nabla ^2 u]$, we consider the tangential derivative of the jump $[\epsilon \nabla u\cdot {\textbf{n}}]$ along ${\textbf{s}}_m$, which satisfies

$$\begin{aligned} \nabla [\epsilon \nabla u\cdot {\textbf{n}}]\cdot {\textbf{s}}_m = \nabla \sigma \cdot {\textbf{s}}_m, \end{aligned}$$

where $\sigma$ is the jump of the flux. Expanding, we can get

$$\begin{aligned} \nabla \left[ \epsilon \nabla u \cdot {\textbf{n}}\right] \cdot {\textbf{s}}_m= & {} {\textbf{s}}_m^T \nabla (\epsilon ^+\left[ \nabla u\right] + \left[ \epsilon \right] \nabla u^-) {\textbf{n}} + {\textbf{s}}_m^T \nabla {\textbf{n}} (\epsilon ^+\left[ \nabla u\right] + \left[ \epsilon \right] \nabla u^-)\\= & {} \epsilon ^+{\textbf{s}}_m^T \left[ \nabla ^2u\right] {\textbf{n}} + {\textbf{s}}_m^T \nabla \epsilon ^+[\nabla u] \cdot {\textbf{n}} + \left[ \epsilon \right] {\textbf{s}}_m^T \nabla ^2u^- {\textbf{n}} + {\textbf{s}}_m^T \left[ \nabla \epsilon \right] \nabla u^- \cdot {\textbf{n}} \\{} & {} + \epsilon ^+{\textbf{s}}_m^T \nabla {\textbf{n}} \left[ \nabla u\right] + \left[ \epsilon \right] {\textbf{s}}_m^T \nabla {\textbf{n}} \nabla u^-. \end{aligned}$$

Now, substituting $[\nabla u]$ by equation (11) and rearranging, we get (20):

$$\begin{aligned} \begin{aligned} {\textbf{s}}_m^T [\nabla ^2u]{\textbf{n}}&= \frac{1}{\epsilon ^+} \nabla \sigma \cdot {\textbf{s}}_m - \frac{[\epsilon ]}{\epsilon ^+} {\textbf{s}}_m^T \nabla ^2u^- {\textbf{n}}- \frac{[\epsilon ]}{\epsilon ^+} {\textbf{s}}_m^T \nabla {\textbf{n}}\nabla u^- \\&- \sum _{k=1}^{d-1} (\nabla \tau \cdot {\textbf{s}}_k) {\textbf{s}}_m^T \nabla {\textbf{n}}{\textbf{s}}_k - \frac{1}{(\epsilon ^+)^2} (\nabla \epsilon ^+ \cdot {\textbf{s}}_m ) ( \sigma - [\epsilon ] \nabla u^- \cdot {\textbf{n}})\\&- \frac{1}{\epsilon ^+} [\nabla \epsilon \cdot {\textbf{s}}_m] (\nabla u^- \cdot {\textbf{n}}). \end{aligned} \end{aligned}$$

Jump of PDE: For our final equations on terms of $[\nabla ^2 u]$, we consider the original PDE. In $\varOmega ^+$, we have

$$\begin{aligned} -\epsilon ^+\varDelta u^+ - \nabla \epsilon ^+\cdot \nabla u^+ + a^+ u^+ = f^+, \end{aligned}$$

while in $\varOmega ^-$, we have

$$\begin{aligned} -\epsilon ^-\varDelta u^- - \nabla \epsilon ^-\cdot \nabla u^- + a^- u^- = f^-. \end{aligned}$$

Dividing these equations by $\epsilon ^+$ and $\epsilon ^-$, respectively, and finding the jump from their difference, we get

$$\begin{aligned}{}[\varDelta u]= & {} -\left[ \frac{f}{\epsilon }\right] - \left[ \frac{\nabla \epsilon }{\epsilon }\cdot \nabla u\right] + \left[ \frac{a}{\epsilon }u\right] \\= & {} -\left[ \frac{f}{\epsilon }\right] -\frac{\nabla \epsilon ^+}{\epsilon ^+} \cdot \left[ \nabla u\right] -\left[ \frac{\nabla \epsilon }{\epsilon }\right] \cdot \nabla u^- +\frac{a^+}{\epsilon ^+}\left[ u\right] +\left[ \frac{a}{\epsilon }\right] u^-\\= & {} -\left[ \frac{f}{\epsilon }\right] -\left[ \frac{\nabla \epsilon }{\epsilon }\right] \cdot \nabla u^- + \frac{a^+}{\epsilon ^+}\left[ u\right] + \left[ \frac{a}{\epsilon }\right] u^-\\{} & {} -\frac{1}{\epsilon ^+}\left( \left[ \nabla u \cdot {\textbf{n}}\right] (\nabla \epsilon ^+\cdot {\textbf{n}}) +\sum _{j=1}^{d-1}(\nabla \tau \cdot {\textbf{s}}_j) (\nabla \epsilon ^+\cdot {\textbf{s}}_j) \right) , \end{aligned}$$

which is (22).

The left hand side of equations (18) (20) and (22):

$$\begin{aligned} {\textbf{s}}_n^T [\nabla ^2u]{\textbf{s}}_m,\quad {\textbf{s}}_m^T [\nabla ^2u]{\textbf{n}}, \quad [\varDelta u] \quad \text {for}\ m = 1,\dots ,d-1\ \text {and}\ n = m,\dots ,d-1 \end{aligned}$$

where $[\nabla ^2u]$ are the unknown quantities, can be written as in the form of a matrix–vector product

$$\begin{aligned} G \left( \left[ \frac{\partial ^{2} u}{\partial x_k \partial x_l}\right] \right) _{1 \le k \le l \le d}. \end{aligned}$$

where G is a matrix that only depends on the normal and the tangent vectors, and the vector is a half-vectorization of the jump of the symmetric Hessian matrix $[\nabla ^2u]$.

We can show that the absolute value of the determinant of G is 1 in two and three dimensions. Since the equations are obtained at some interface point ${\hat{\textbf{x}}}$, we can use a local coordinate system such that ${\textbf{s}}_i = {\textbf{e}}_i$ for $i=1,\dots ,d-1$ and ${\textbf{n}} = {\textbf{e}}_d$. By choosing a specific ordering for the equations and the half-vectorization, we can write the matrix–vector product in 2D as

$$\begin{aligned} \begin{pmatrix} {\textbf{s}}_1 [\nabla ^2u]{\textbf{s}}_1\\ {\textbf{s}}_1 [\nabla ^2u] {\textbf{n}} \\ [\varDelta u] \end{pmatrix} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 1 &{} \quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1\\ \end{pmatrix} \begin{pmatrix} [u_{xx}]\\ [u_{xy}]\\ [u_{yy}]\\ \end{pmatrix} \end{aligned}$$

And in 3D

$$\begin{aligned} \begin{pmatrix} {\textbf{s}}_1 [\nabla ^2u]{\textbf{s}}_1\\ {\textbf{s}}_2 [\nabla ^2u]{\textbf{s}}_2\\ [\varDelta u]\\ {\textbf{s}}_1 [\nabla ^2u] {\textbf{s}}_2\\ {\textbf{s}}_1 [\nabla ^2u] {\textbf{n}}\\ {\textbf{s}}_2 [\nabla ^2u] {\textbf{n}}\\ \end{pmatrix} = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{} \quad 0 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{} \quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ \end{pmatrix} \begin{pmatrix} [u_{xx}]\\ [u_{xy}]\\ [u_{yy}]\\ [u_{xy}]\\ [u_{xz}]\\ [u_{yz}]\\ \end{pmatrix} \end{aligned}$$

Therefore, the determinant of G is $\pm 1$, depending on the ordering of the equations and the half-vectorization.

High Contrast Problem

One difficulty of interface problems is the so-called large contrast problem, where the ratio of the coefficients $\epsilon ^+/\epsilon ^-\gg 1$. There are several works that analyze the high-contrast problems in the context of unfitted Nitsche finite element method [71] and unfitted finite element methods [38, 43, 71, 72]. For example, it can be shown that the flux error estimate is independent of the contrast for a class of unfitted Nitsche finite element methods [71].

Here we numerically demonstrate that our method is robust for high contrast problems. We consider the same exact solution (36) and coefficients (37) as in Example 1, but with $\epsilon ^-=1$ and $\epsilon ^+=1$ or $\epsilon ^+=1e6$. Figure 16 shows the convergence result of the six interfaces. We see that both the solution and the gradient at the interface are uniformly second-order accurate for both cases. We also see that the error between the two cases is similar, demonstrating the robustness of our method for high contrast problems. The theoretical analysis of the high contrast problem is beyond the scope of this paper and will be studied in the future.

Fig. 16

The log–log plot of the error versus N for the six surfaces. $\epsilon ^-=1$ and $\epsilon ^+=1$ or $\epsilon ^+=1e6$. In each figure, N ranges from 50 to 140 with the increment $\varDelta N = 5$. “Sol” denotes maximum errors of the solution $\left\| u_e-u\right\| _\infty$. “Grad” denotes the maximum errors of the gradient at interface $\left\| \nabla u_e - \nabla u\right\| _{\infty ,\varGamma }$. m is the slope of the fitting line