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The main question is: How to deal with the Poisson equation in the presence of the medium interface.

Let's say we have 1D Laplace equation: \begin{equation} -\frac{d}{dx}\left(\epsilon(x)\frac{d}{dx}\right)\phi(x)=0 \end{equation} or \begin{equation} -\left(\frac{d\epsilon}{dx}\frac{d}{dx}+\epsilon\frac{d^2}{dx^2} \right)\phi(x)=0. \end{equation} Now, consider the case when \begin{equation} \epsilon(x) = \begin{cases} \epsilon_1 & \mbox{if } x<0 \\ \epsilon_2, & \mbox{if } x>0 \end{cases} \end{equation}.

If I put the equation on the equally spaced grid \begin{equation} x=n\cdot a, \ \ n=...-2,-1,0,1,2... \end{equation} ($a$ is the discretization constant) and replace the continues derivatives with the discrete counterparts, there is a problem at $x=0$ with the \begin{equation} \frac{d\epsilon}{dx}\approx \frac{\epsilon(x+a)-\epsilon(x-a)}{2a}, \end{equation} because it's clearly depend on the grid spacing. Even in the limit $a\rightarrow0 $ there seems to be a problem. What is the typical method to deal with such a problem?

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A surface charge will accumulate on the interface where the dielectric constant has a discontinuity. You need to calculate this surface charge and include it into the discretised Poisson equation. Here are two papers which show how this is done in practice (there is an extensive literature, these are just two pointers to get started):

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