Edge-face Maxwell finite element semidiscretization

Let $\mathrm{\Omega }$ be a bounded contractible Lipschitz polyhedron with a finite matching affine tetrahedral mesh ${\mathcal{T}}_h$, and use the real lowest-order spaces $W_h^0,N_h^1,N_h^2,N_h^3$ defined in Lowest-order Whitney and Nedelec de Rham spaces. The electric trial field is $E_h(t)\in N_h^1\cap H_0(\mathrm{curl};\mathrm{\Omega })$; the magnetic flux field is $B_h(t)\in N_h^2$, with no imposed zero normal trace on $B_h$. PEC means the closure-defined zero tangential subspace from Zero tangential trace H(curl) space. Periodic analogues use the periodic cell complex. Countable Choice is inherited from the H(div) and affine Piola space definitions in the suppliers; the finite matrix operations below use no additional choice.

Put $V_h:=N_h^1\cap H_0(\mathrm{curl};\mathrm{\Omega })$. Choose an edge basis $\{{\varphi }_i{\}}_{i=1}^{N_e}$ of the electric trial space $V_h$ and a face basis $\{{\psi }_a{\}}_{a=1}^{N_f}$ of $N_h^2$, so $N_e=\dim V_h$ and $N_f=\dim N_h^2$. Define the symmetric mass matrices $(M_e{)}_{ij}={\int }_{\mathrm{\Omega }}{\varphi }_j^Tϵ{\varphi }_{i}dx,(M_b{)}_{ab}={\int }_{\mathrm{\Omega }}{\psi }_b^T{\mu }^{-1}{\psi }_{a}dx\mathrm{.}$ They are positive definite by the coefficient bounds in Transient Maxwell system with piecewise materials and boundary conditions and linear independence of the finite element bases. Let $C$ and $D$ be the coefficient matrices, in these chosen bases, of the restricted map $\mathrm{curl}:V_h\to N_h^2$ and of $\mathrm{div}:N_h^2\to N_h^3$. The moment-commuting supplier gives $DC=0$ on the full spaces, hence also after restriction to $V_h$. Set $j_i(t)={\int }_{\mathrm{\Omega }}{J}_{\mathrm{s}\mathrm{r}\mathrm{c}}(t)\cdot {\varphi }_{i}dx$, with $J_{\mathrm{s}\mathrm{r}\mathrm{c}}\in C([0,T];L^2(\mathrm{\Omega };{\mathbb{R}}^3))$. The lossless, conductivity-free semidiscretization with this source is $M_e\dot{e}-C^T{M}_{b}b=-j,\dot{b}+Ce=0.$ The coefficient ODE is defined for arbitrary initial coefficients. To represent the magnetic constraint of the Maxwell model, require $Db(0)=0$: the Faraday row and $DC=0$ give $D\dot{b}=-DCe=0$, so $Db(t)=Db(0)$ and compatible data remain divergence-free. An incompatible initial divergence is preserved, not corrected by this ODE. This is a spatial Galerkin system only; no time integrator is included. The cochain convention and weak material-interface formulation come from the declared spaces and Transient Maxwell system with piecewise materials and boundary conditions. Smooth interpolation and a uniformly bounded commuting projection are separate prerequisites for the later error estimate and are not asserted by this definition.