Transient Maxwell system with piecewise materials and boundary conditions

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded Lipschitz polyhedron, or a periodic rectangular cell. The real electric field is $E(t,x)\in {\mathbb{R}}^3$ and the magnetic flux density is $B(t,x)\in {\mathbb{R}}^3$; the magnetic intensity is $H={\mu }^{-1}B$. Let $ϵ(x)$ and $\mu (x)$ be measurable, time-independent, symmetric matrix fields for which there are constants $0<{ϵ}_0\le {ϵ}_1$ and $0<{\mu }_0\le {\mu }_1$ with ${ϵ}_0\mathrm{\mid }v{\mathrm{\mid }}^2\le v^Tϵ(x)v\le {ϵ}_1\mathrm{\mid }v{\mathrm{\mid }}^2,{\mu }_0\mathrm{\mid }v{\mathrm{\mid }}^2\le v^T\mu (x)v\le {\mu }_1\mathrm{\mid }v{\mathrm{\mid }}^2$ for almost every $x$ and every $v\in {\mathbb{R}}^3$. A conductivity tensor $\sigma (x)$, when present, is measurable, symmetric, and positive semidefinite. The coefficients may jump across finitely many Lipschitz material interfaces. No derivative of a coefficient is part of the model.

In each material region the constitutive fields satisfy ${\mathrm{\partial }}_t(ϵE)=\mathrm{curl}({\mu }^{-1}B)-J_{\mathrm{s}\mathrm{r}\mathrm{c}}-\sigma E,{\mathrm{\partial }}_{t}B+\mathrm{curl}E=0.$ Here $J_{\mathrm{s}\mathrm{r}\mathrm{c}}$ is the impressed free-current density, while $\sigma E$ is the conductive current. Electric charge $\rho$ is constrained by $\mathrm{div}(ϵE)=\rho$ and the total-current continuity equation ${\mathrm{\partial }}_t\rho +\mathrm{div}(J_{\mathrm{s}\mathrm{r}\mathrm{c}}+\sigma E)=0$; the magnetic constraint is $\mathrm{div}B=0$. These equalities are distributional. In the lossless branch set $\sigma =0$.

The boundary choice is one of the following. A periodic boundary identifies opposite traces of the rectangular cell. A perfect electric conductor (PEC) boundary imposes zero tangential trace $n\times E=0$ on $\mathrm{\partial }\mathrm{\Omega }$. At a material interface with no singular surface charge or current, the weak equations are understood with the transmission conditions $[n\times E]=0$, $[n\cdot B]=0$, $[n\times ({\mu }^{-1}B)]=0$, and $[n\cdot (ϵE)]=0$, where $[q]$ is the trace from one side minus the trace from the other. The interface conditions are part of the model convention; no smooth Taylor expansion across a coefficient jump is asserted.

For PEC, the weak Ampere equation is $(ϵE_t,v{)}_{L^2}-({\mu }^{-1}B,\mathrm{curl}v{)}_{L^2}=-(J_{\mathrm{s}\mathrm{r}\mathrm{c}}+\sigma E,v{)}_{L^2}$ for every $v\in H_0(\mathrm{curl};\mathrm{\Omega })$ as defined in Zero tangential trace H(curl) space; the periodic branch uses periodic $H(\mathrm{curl})$ test fields. Faraday’s equation remains $B_t+\mathrm{curl}E=0$ in distributions. These weak forms define the test-space and boundary convention used by the edge finite element method.

The weighted field norms are $\mathrm{\parallel }E{\mathrm{\parallel }}_{ϵ}^2={\int }_{\mathrm{\Omega }}{E}^TϵEdx$ and $\mathrm{\parallel }B{\mathrm{\parallel }}_{{\mu }^{-1}}^2={\int }_{\mathrm{\Omega }}{B}^T{\mu }^{-1}Bdx$. The uniform coefficient bounds make them equivalent to the ordinary $L^2$ norms. All later smooth-error claims require smoothness separately on each mesh-aligned material region.