Abstract
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.
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Notes
- 1.
- 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.
If this is not the case, we refer to Sect. 6.3.
- 4.
See footnote 3, p. 291.
- 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.
- 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.
References
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P. Ciarlet, Jr., S. Labrunie, Numerical analysis of the generalized Maxwell equations (with an elliptic correction) for charged particle simulations. Math. Models Methods Appl. Sci. 19, 1959–1994 (2009)
P. Ciarlet, Jr., S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier Singular Complement Method. Differ. Equ. Appl. 3, 113–155 (2011)
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Assous, F., Ciarlet, P., Labrunie, S. (2018). Analyses of Exact Problems: Second-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_7
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