## The Boundary Integral Equation

The so-called **boundary integral equation** relates the values of the electrostatic potential in some domain to its values at that domain's boundary. In this problem we will **derive** this important statement which leads to the "**Boundary Element Method**", a discretized version with **numerical applications** throughout science and engineering.

## Problem Statement

Derive the boundary integral equation for a region \(\Omega\) containing no charges:

- Given the Poisson equation for a function \(u\left(\mathbf{r}\right)\), \(\Delta u\left(\mathbf{r}\right)=0\) in some domain \(\Omega\) and likewise for \(w\left(\mathbf{r}\right)\), derive
**Green's second identity**\[\int_{\Omega}u\left(\mathbf{r}\right)\Delta w\left(\mathbf{r}\right)dV-\int_{\Omega}w\left(\mathbf{r}\right)\Delta v\left(\mathbf{r}\right)dV = \int_{\partial\Omega}\left[u\left(\mathbf{r}\right)\partial_{\boldsymbol{\mathbf{\nu}}}w\left(\mathbf{r}\right)-w\partial_{\boldsymbol{\mathbf{\nu}}}u\left(\mathbf{r}\right)\right]d\mathbf{A}\ .\] - Now
**derive**the actual**boundary integral equation**: Green's function for the Laplace operator is defined by \(\Delta G\left(\mathbf{r},\mathbf{r}^{\prime}\right)=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\). Apply Green's second identity now for the electrostatic potential \(\phi\left(\mathbf{r}\right)\) and \(G\left(\mathbf{r},\mathbf{r}^{\prime}\right)\). - Regarding the boundary integral equation, can you imagine what kind of
**problem**may arise if one uses it in a**numerical implementation**?

Background: The Boundary Element Method

Using Green's second identity, the boundary integral equation relates the values of the electrostatic potential inside some domain \(\Omega\) to its values at the boundary of \(\Omega\). This is per se not really useful because we might be able to study just a single dielectric body in some embedding medium. However, the approach can be **generalized** to several domains, i.e. more complex bodies - **examples** for such bodies can be seen on the left: a) a **photonic crystal** embedded in some metal, b) a (much) more **sophisticated structure**.

The generalization is done using the appropriate **boundary conditions** between adjacent domains. This leads to a system of coupled boundary integral equations which can be **discretized** replacing the integrals by sums. In turn, the system of boundary integral equations can be evaluated on a computer - the numerical approach is called “**Boundary Element Method**”. It has a lot of applications in optics, acoustics, aerodynamics since one can use this this method also for other kinds of wave equations. Within the Boundary Element Method, several scientific and engineering questions can be cast into questions on the arising system of equations, hence properties of some matrix. For instance, one can calculate the **response** of a dielectric body to a giving excitation or a direct computation of the **resonances**. A lot of books and papers exist. For a realization with an optics/nanophotonics background, see J. Wierig's excellent “Boundary element method for resonances in dielectric microcavities”, Journal of Optics A: Pure and Applied Optics **5**, 2003.

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