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Solving Maxwell's Equations in time-harmonic form.

The equations solved are the time-harmonic Maxwell's equations:

$- \varepsilon i \omega \vec E + \text{curl} \vec H = \sigma \vec E + \vec J$

$\mu i \omega \vec H + \text{curl} \vec E = 0$

with the divergence conditions (assuming no charge density)

$\text{div} \mu \vec H = 0 \qquad \text{and} \qquad \text{div} \varepsilon \vec E = 0.$

Extracting $\vec H$ from the second equation and inserting in into the first formally yields

$\text{curl}( \mu^{-1} \text{curl} \vec E) - \omega^2 (\varepsilon + \frac{\sigma}{i\omega}) \vec E = -i \omega \vec J,$

the so-called electric source problem. The appropriate space is

$H(\text{curl}; D) := \{\vec u \in L^2(D)^3 : \text{curl} \vec u \in L^2(D)^3 \}.$

However, the curl-curl form above is not defined for electric fields in $H(\text{curl}; D)$ . This is resolved in the variational formulation of the electric source problem:
Find $\vec E \in H_0(\text{curl}; D)$ such that

$\int_D (\mu^{-1} \text{curl} \vec E \cdot \text{curl} \vec v - \omega^2 \tilde\varepsilon \vec E \cdot \vec v) \, d\vec x = \int_D \vec f \cdot \vec v \, d\vec x \quad \forall \vec v \in H_0(\text{curl}; D),$

where $H_0(\text{curl}; D)$ is $H(\text{curl}; D)$ with the prefect conductor boundary conditions $\vec E \times \vec n = 0$ , $\tilde \varepsilon := \varepsilon + \frac{\sigma}{i\omega}$ and $f = -i\omega \vec J$ . The associated bilinear operator of this variational formulation is not elliptic and the divergence condition is an independent constraint. Normally, Maxwell's equations are discretised using Nedelec's elements. However, there are good reasons why one would like to use standard H¹-conforming FEM. This is possible using weighted regularization.

See also: Two dimensions: Edge elements., Weighted regularization; Three dimensions: Weighted regularization

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