4. The Scalar Finite Element Method

One typical application of the scalar finite element method is the numerical solution of a Poisson equation. Thus it is used for a detailed explanation of the basic concept of this method.

The Poisson equation is derived from the Maxwell equations [41,42] for the static case -- the field quantities do not vary with time. The differential form of the four Maxwell equations is usually given as

$$\vec{\nabla}\times\vec{E} = -\partial_t\vec{B} \mathrm{(Faraday's law)}$$ (4.1)

$$\vec{\nabla}\cdot\vec{B} = 0 \mathrm{(Gau{\ss}'s law for magnetism)}$$ (4.2)

$$\vec{\nabla}\times\vec{H} = \vec{J} + \partial_t\vec{D} \mathrm{(Maxwell{-}Amp\grave{e}re law)}$$ (4.3)

$$\vec{\nabla}\cdot\vec{D} = \rho \mathrm{(Gau{\ss}'s law)},$$ (4.4)

where each variable has the following meaning and unit:

$$\renewedcommand{arraystretch}{1.3} \begin{array}{lll} \vec{E} & \... ...thrm{electric charge density} & \frac{\mathrm{As}}{\mathrm{m}^3}. \end{array}$$

In this work $\vec{E}$ and $\vec{H}$ will be also referred to as electric field and magnetic field, respectively. Applying the divergence operator to (4.3) and substitution by (4.4) give

$$\vec{\nabla}\cdot\vec{J} = -\partial_t\rho \mathrm{(equation of continuity).}$$ (4.5)

The macroscopic properties of the medium are described in terms of permittivity $\epsilon$ , permeability $\mu$ , and conductivity $\gamma$ . These parameters are used for specifying the constitutive relations between the field quantities

$$\vec{D} = \epsilon\vec{E}$$ (4.6)

$$\vec{B} = \mu\vec{H}$$ (4.7)

$$\vec{J} = \gamma\vec{E}.$$ (4.8)

In general it is not necessary that the constitutive parameters $\epsilon$ , $\mu$ , and $\gamma$ are simple constants. For example the relationship between $\vec{B}$ and $\vec{H}$ in (4.7) may by highly non-linear for ferromagnetic materials. In this case $\mu$ depends on the field. For anisotropic media the directions of the flux densities differ from the directions of the corresponding field intensities and the constitutive parameters must be described by tensors. In inhomogeneous regions $\epsilon$ , $\mu$ , and $\gamma$ are functions of position.

If the field values are invariant in time, the field is static. In this case the magnetic field and the electric field do not interact. For instance, the electrostatic case is given by (4.4) and

$$\vec{\nabla}\times\vec{E} = 0.$$ (4.9)

Equation (4.9) is satisfied by

$$\vec{E} = -\vec{\nabla}\varphi.$$ (4.10)

After substituting (4.10) in (4.6) and insertion in (4.4), the following second order partial differential equation for the electrostatic potential $\varphi$ is obtained

$$\vec{\nabla}\cdot\left[\utilde{\epsilon}(\vec{r})\cdot \vec{\nabla}\varphi(\vec{r})\right] = f(\vec{r}), \mathrm{with} f(\vec{r}) = -\rho.$$ (4.11)

If assumed that $\utilde{\epsilon}(\vec{r})$ is a constant scalar, the expression (4.11) turns into the well known Poisson equation. If the charge density is zero allover the domain, (4.11) leads to

$$\vec{\nabla}\cdot\left[\utilde{\epsilon}(\vec{r})\cdot \vec{\nabla}\varphi(\vec{r})\right] = 0,$$ (4.12)

which corresponds for constant scalar permittivity $\epsilon$ to the Laplace equation. The most general presentation for the permittivity, which can be handled by the finite element method in the frequency domain is a position dependent tensor $\utilde{\epsilon}(\vec{r})$ .