Abstract
In this chapter, we devote our attention to establishing mathematical properties concerning the electromagnetic fields that are governed by the time-dependent Maxwell equations. For that, we investigate a number of physical properties of the electromagnetic fields exhibited in Chap. 1, using the mathematical tools introduced in Chaps. 2, 3 and 4. We focus mainly on four items.
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Notes
- 1.
See Remark 5.1.2 on how to take into account the boundary condition on the magnetic field.
- 2.
Since one can choose where to put the artificial boundary Γ A , it is a reasonable assumption. Also, because Γ A is smooth, one has
$$\displaystyle \begin{aligned} \boldsymbol{H}^{1/2}_\parallel(\varGamma_A)=\boldsymbol{H}^{1/2}_\perp(\varGamma_A)=\boldsymbol{H}^{1/2}_t(\varGamma_A),\mbox{ where }\boldsymbol{H}^{1/2}_t(\varGamma_A):=\boldsymbol{L}^{2}_t(\varGamma_A)\cap\boldsymbol{H}^{1/2}(\varGamma_A), \end{aligned}$$and similarly for the dual spaces, \(\boldsymbol {H}^{-1/2}_\parallel (\varGamma _A)=\boldsymbol {H}^{-1/2}_\perp (\varGamma _A)=\boldsymbol {H}^{-1/2}_t(\varGamma _A)\).
- 3.
If ∂Γ P ∩ ∂Γ A = ∅, one still needs to address the possible lack of regularity of the artificial boundary (see Remark 5.1.6). This corresponds to configurations 2 and 3 of Γ A in the study below.
- 4.
One applies the Lax-Milgram Theorem 4.2.8 to the equivalent variational form:
If Γ A is not a connected set, one chooses—instead of \(H^1_0(\varGamma _A)\)—the space
$$\displaystyle \begin{aligned} \{ f\in H^1(\varGamma_A)\ :\ f_{\mid{\partial\varGamma_A^0}}=0,\ f_{\mid{\partial\varGamma_A^k}}=cst_k,\ 1\le k\le K_A\}, \end{aligned}$$where \((\varGamma _A^k)_{k=0,K_A}\) are the (maximal) connected components of Γ A .
- 5.
For \( \underline \psi = \underline \psi ^++\sqrt {{\mu }/{\varepsilon }} \underline \psi ^-\), we have: \( \underline \psi \in H^{1/2}(\varGamma _A)\), \(\varDelta _\varGamma \underline \psi =0\) in Γ A , \(t_{\boldsymbol {\nu }}( \operatorname {\mathrm {\mathbf {grad}}}_\varGamma \underline \psi )=0\). In this case, we are looking for singular solutions (with at least H 1∕2-regularity) to the Laplace-Beltrami problem with homogeneous Neumann boundary condition and right-hand side. Completely similar analyses can be carried out for \( \underline \psi \): they yield the same results as for \( \underline \phi \).
- 6.
More precisely, we recall that, if \( \underline \phi \) belongs to H 1(Γ A ) with \( \underline \phi { }_{\mid {\partial \varGamma _A}}=0\), we apply the integration by parts (5.23) to find that
. If one replaces \( \underline \phi \) with \( \underline \phi '= \underline \phi +c\) with c≠0, then the technique still applies (even though \( \underline \phi '{ }_{\mid {\partial \varGamma _A}}\ne 0\)), because \(\varDelta _\varGamma \underline \phi '=0\). So, the local constant behavior can be neglected. - 7.
See footnote 5, p. 12.
- 8.
- 9.
- 10.
The situation is different for time-harmonic problems (see Chap. 8).
. If one replaces References
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Assous, F., Ciarlet, P., Labrunie, S. (2018). Analyses of Exact Problems: First-Order Models. In: Mathematical Foundations of Computational Electromagnetism. Applied Mathematical Sciences, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-70842-3_5
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