3.1 Boundary-Value Problems

The boundary-value problems can be described by the expression

$$\mathcal{L}\left[u\left(\vec{r}\right)\right] = f(\vec{r})$$ (3.1)

in a domain $\mathcal{V}$ and by the boundary conditions applied on the boundary $\partial\mathcal{V}$ enclosing the domain $\mathcal{V}$ . In (3.1) $\mathcal{L}$ is a differential operator, $u$ is the unknown function, and $f$ is the excitation or forcing term. The differential equations used in this thesis are linear. Therefore, $\mathcal{L}$ is assumed to be a linear differential operator.

Depending on the type of fields to be solved there are different kinds of linear differential operators $\mathcal{L}$ . In this thesis the linear differential operator appears as scalar operator

$$\mathcal{L}[u] = \vec{\nabla}\cdot(\utilde{a}\cdot\vec{\nabla}u) + bu$$ (3.2)

or as vector operator

$$\vec{\mathcal{L}}[\vec{u}] = \vec{\nabla}\times\left[\utilde{a}\cdot(\vec{\nabla}\times\vec{u})\right] + \utilde{b}\cdot\vec{u}.$$ (3.3)

In (3.2) and (3.3) $\utilde{a} = \utilde{a}(\vec{r})$ and $\utilde{b} = \utilde{b}(\vec{r})$ are position dependent second order symmetric tensors and $b = b(\vec{r})$ is a position dependent scalar factor. The operator $\vec{\nabla}$ is the in the vector analysis widely used differential operator nabla. In Cartesian coordinates in the three-dimensional space $\vec{\nabla}$ has the following representation:

$$\vec{\nabla} = \vec{e}_x\partial_{x} + \vec{e}_y\partial_{y} + \vec{e}_z\partial_{z}.$$

The partial derivative, for example with respect to the variable $x$ , is abbreviated as $\partial_x$ instead of $\frac{\partial}{\partial{}x}$ . The expressions (3.2) and (3.3) represent linear differential operators. In (3.2) $\mathcal{L}$ is a scalar differential operator which operates on the scalar function $u$ . In (3.3) $\vec{\mathcal{L}}$ is a vector differential operator which operates on the vector function $\vec{u}$ .

The boundary $\partial\mathcal{V}$ is divided as in [35] into a Dirichlet boundary $\mathcal{A}_D$ and a Neumann boundary $\mathcal{A}_N$ , where

$$\partial\mathcal{V} = \mathcal{A}_D + \mathcal{A}_N.$$

For elliptical partial differential equations, like the Helmholtz equation $\vec{\nabla}\cdot\vec{\nabla}u + bu = f(\vec{r})$ , the Dirichlet boundary condition (or the boundary condition of the first kind)

$$u(\vec{r}) = u_d \mathrm{on} \mathcal{A}_D$$

is given by the values of $u$ on the Dirichlet boundary $\mathcal{A}_D$ and the Neumann boundary condition (or the boundary condition of the second kind)

$$\vec{n}\cdot\vec{\nabla}u(\vec{r}) = u_n \mathrm{on} \mathcal{A}_N.$$

by the normal derivative of $u$ on the Neumann boundary $\mathcal{A}_N$ , respectively [36]. For the more general case (3.2) the Neumann boundary condition has to be specified by the conormal derivative

$$\vec{n}\cdot\utilde{a}\cdot\vec{\nabla}u(\vec{r}) = u_n \mathrm{on} \mathcal{A}_N.$$ (3.4)

instead of the normal derivative [37].

The same can be written for two-dimensional problems. In this case the domain is an area denoted as $\mathcal{A}$ with the boundary $\partial\mathcal{A}$

$$\partial\mathcal{A} = \mathcal{C}_D + \mathcal{C}_N.$$