Oriented Maxwell cochains, masses, and the Yee grid

Let $K$ be a finite oriented three-dimensional cell complex with vertex, edge, face, and volume cochains. An edge cochain records oriented electric line integrals, a face cochain records oriented magnetic fluxes, and a vertex cochain records electric charge. Write $N_v,N_e,N_f,N_c$ for the retained vertex, edge, face, and cell counts. On a periodic cell, identify opposite cells with orientation preserved and let $G,C,D$ be the full vertex-to-edge gradient, edge-to-face curl, and face-to-cell divergence incidence matrices, each the transpose of its signed oriented boundary matrix. For a PEC closure, “interior” means not in the boundary subcomplex. Keep only interior electric edges as unknowns and only interior vertices as volume-charge locations, but keep all face fluxes, including boundary-normal magnetic fluxes, and all cells. This closure has no separate boundary surface-charge cochain. Define $G$ and $C$ by restricting the absolute vertex-edge and edge-face incidence matrices to those rows and columns. This mixed closure imposes zero tangential electric edge cochains on the boundary while allowing a fixed normal magnetic flux on boundary faces. In both cases, $G\in {\mathbb{R}}^{N_e\times N_v}$, $C\in {\mathbb{R}}^{N_f\times N_e}$, and $D\in {\mathbb{R}}^{N_c\times N_f}$ satisfy $CG=0$ and $DC=0$.

The electric, magnetic, and conductive material matrices are real symmetric matrices $M_e\in {\mathbb{R}}^{N_e\times N_e}$, $M_b\in {\mathbb{R}}^{N_f\times N_f}$, and $S\in {\mathbb{R}}^{N_e\times N_e}$, respectively. The first two are positive definite and represent the discrete weighted inner products induced by $ϵ$ and ${\mu }^{-1}$; $S$ is positive semidefinite and represents conductivity. Write $⟨e_1,e_2{⟩}_{M_e}=e_1^T{M}_e{e}_2$, $⟨b_1,b_2{⟩}_{M_b}=b_1^T{M}_b{b}_2$, and $\mathrm{\parallel }e{\mathrm{\parallel }}_{M_e}^2=⟨e,e{⟩}_{M_e}$, $\mathrm{\parallel }b{\mathrm{\parallel }}_{M_b}^2=⟨b,b{⟩}_{M_b}$. In a homogeneous isotropic periodic Cartesian grid with spacings $h_x,h_y,h_z$ and cell volume $V=h_x{h}_y{h}_z$, the diagonal entries for an $x$-edge and an $x$-normal face are $ϵV\mathrm{/}{h}_x^2=ϵh_y{h}_z\mathrm{/}{h}_x$ and ${\mu }^{-1}{h}_x\mathrm{/}(h_y{h}_z)$; the $y$ and $z$ entries are obtained by cyclic permutation. These values are dual-volume Hodge weights for line-integral and face-flux cochains. A compatible PEC closure uses the corresponding truncated dual volumes for free electric edges, while the magnetic mass includes boundary faces; the resulting matrices remain positive definite. The discrete constraints are $-G^T{M}_e{e}^n={\rho }^n$ on interior vertices and $D{b}^{n-1\mathrm{/}2}=0$ on all cells.

The minus sign in the electric constraint is essential: an edge oriented from tail to head has incidence entries $-1,+1$, so $-G^T$ is the outgoing-flux divergence. Thus $\rho$ represents physical electric charge with the sign of $\mathrm{div}(ϵE)$ in Transient Maxwell system with piecewise materials and boundary conditions.

For current cochains $j^{n+1\mathrm{/}2}$ and charge rows ${\rho }^n$, the discrete continuity condition means ${\rho }^{n+1}-{\rho }^n-\mathrm{\Delta }t{G}^T(j^{n+1\mathrm{/}2}+S\frac{e^{n+1}+e^n}{2})=0.$ For PEC this is the interior-vertex volume-charge relation; this closure has no separate boundary surface-charge/current cochain.

The cochain identities $CG=0$ and $DC=0$ are not imposed as an extra numerical approximation: they follow from oriented boundary-of-boundary cancellation, including the PEC row and column restrictions, as proved in Incidence curl has zero discrete divergence and gradient curl ↗. No cell geometry picture is used as a proof.