Analyses of Exact Problems: Second-Order Models

This chapter is devoted to an alternative, second-order formulation of the Maxwell’s equations. We rigorously justify the process we outlined in Sect. 1.5.3. This new formulation is especially relevant for computational applications, as it admits several variational formulations, which can be simulated by versatile finite element methods. Our attention will be focused on three issues: equivalence of the second-order equations with the original, first-order equations studied in Chap. 5, the well-posedness of the new formulation and the regularity of its solution, as we did in that chapter. We also study how to take into account the conditions on the divergence of the fields, incorporating them explicitly at some point in the variational formulations. To these ends, we shall again rely on the mathematical tools introduced in Chaps. 2, 3 and 4, as well as on the specific properties of the spaces of electromagnetic fields introduced in Chap. 6.

1. 1.

   The other way around (see Sect. 7.3) the conditions on allow one to recover the perfect conductor boundary condition on E by integrating in time, with the help of Proposition 2.3.4.

2. 2.

   From the definition \(\boldsymbol {k}^\star =\sqrt {{\varepsilon }/{\mu }}\,\boldsymbol {g}^\star \times \boldsymbol {n}\), and because g ^⋆ is tangential, it holds that \(\sqrt {{\mu }/{\varepsilon }}\,\boldsymbol {k}^\star \times \boldsymbol {n} = - \boldsymbol {g}^\star \). Hence, the boundary term in the r.h.s. of (7.21) also writes

   $$\displaystyle \begin{aligned} -\int_0^T {{}_{\gamma_A^0}}\langle\boldsymbol{g}^\star,\boldsymbol{w}_{\top}\rangle_{\pi_A}\varphi'(t)\,dt. \end{aligned}$$
3. 3.

   If this is not the case, we refer to Sect. 6.3.

4. 4.

   See footnote 3, p. 291.

5. 5.

   Given g ∈ H ^−s(Ω), let \(g^\sharp \in H^s_0(\varOmega )\) be defined by the condition

   $$\displaystyle \begin{aligned} \langle g^\sharp , g' \rangle_{H^{-s}(\varOmega)} = ( g , g^{\prime})_{H^{-s}(\varOmega)}, \quad \forall g' \in H^{-s}(\varOmega). \end{aligned}$$

   The isomorphism ^♯ reduces to the identity if s = 0.

6. 6.

   See [17, 81, 82]. These conditions are actually more stringent than those originating in the approximation of the right-hand side.

7. 7.

   If is scalar-valued and belongs to W ^{1, ∞}(Ω), a sufficient condition to ensure this is . In both cases, the condition on ϱ implies \(\varrho ^\sharp \in W^{2,p}(0,T;H^1_0(\varOmega ))\), which is needed for the mixed problem.

Authors and Affiliations

1. Department of Mathematics, Ariel University, Ariel, Israel

   Franck Assous

2. ParisTech, ENSTA, Palaiseau, France

   Patrick Ciarlet

3. Université de Lorraine, Institut Élie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

   Simon Labrunie

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