2. The dual cell method for the time-domain Maxwell system

In this section we describe in detail how the dual cell method is applied to the time-domain Maxwell equations.

2.1. Problem setting

The time-domain Maxwell system (we neglect possible sources here) is the problem to find fields \(\mathbf D,\mathbf B,\mathbf H,\mathbf E\) such that

\[\begin{align*} \partial_t \mathbf{D}(t,\mathbf{x})-\mathrm{curl}\mathbf{H}(t,\mathbf{x})&=0,\\ \partial_t \mathbf{B}(t,\mathbf{x})+\mathrm{curl}\mathbf{E}(t,\mathbf{x})&=0.\\ &+\text{b.c., i.c.,} \end{align*}\]

for \(t\in(0,T),\mathbf x\in\Omega\) and some \(T>0\) and a suitable domain \(\Omega\subset\mathbb R^3\). To close the system one also needs the constitutive relations

\[\begin{align*} \mathbf{D} &= {\color{emph1}\varepsilon}\mathbf{E},& \mathbf{B} &= {\color{emph2}\mu}\mathbf{H}, \end{align*}\]

where \(\varepsilon,\mu\) are the permittivity and permeability of the medium in question.

In weak (EH-)formulation, assuming homogeneous boundary conditions \(\mathbf E \times \mathbf n = 0\), the above problem may be rewritten as the problem to find \(\mathbf E,\mathbf H:[0,T]\to H(\mathrm{curl})(\Omega)\) such that

\[\begin{align*} \int_\Omega\varepsilon \partial_t \mathbf{E}(t,\mathbf{x})\mathbf E'(\mathbf x)d\mathbf x-\int_\Omega\mathrm{curl}\mathbf{H}(t,\mathbf{x})\mathbf E'(\mathbf x)d\mathbf x&=0,\\ \int_\Omega\mu\partial_t \mathbf{H}(t,\mathbf{x})\mathbf H'(\mathbf x)d\mathbf x+\int_\Omega\mathbf{E}(t,\mathbf{x})\mathrm{curl}\mathbf H'(\mathbf x)d\mathbf x&=0.\\ &+\text{i.c.,} \end{align*}\]

for all \(\mathbf E',\mathbf H'\in H(\mathrm{curl})(\Omega)\).