PEC Maxwell cavity eigenproblem and full-kernel gauge

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded connected open set whose boundary is locally the graph of a Lipschitz function (a bounded Lipschitz domain). The zero-tangential boundary condition is the perfect-electric- conductor (PEC) condition. We use PEC to name the closure convention below; for a smooth field this is the usual condition $n\times u=0$ on the boundary. Its space is $V:=H_0(\mathrm{curl};\mathrm{\Omega }),$ with the graph norm and closure convention of The real H(curl) graph space on a polyhedron and Zero tangential trace H(curl) space. For a positive constant permittivity $ϵ$ and a positive constant permeability $\mu$, write $(u,v{)}_{ϵ}:=ϵ{\int }_{\mathrm{\Omega }}u\cdot vdx,K:=\{u\in V:\mathrm{curl}u=0\},X:=V\cap K^{{\perp }_{ϵ}}\mathrm{.}$ Thus $X$ removes the entire PEC curl kernel, without identifying that kernel with a scalar gradient range.

For a bounded polyhedral subcase with a finite matching affine tetrahedral mesh ${\mathcal{T}}_h$ as in Matching affine simplicial meshes and shape regularity, let $V_h:=N_h^1\cap V,K_h:=\{v_h\in V_h:\mathrm{curl}{v}_h=0\},X_h:=V_h\cap K_h^{{\perp }_{ϵ}},$ where $N_h^1$ is the lowest-order first-family edge space of Lowest-order Whitney and Nedelec de Rham spaces. The notation for $K_h$ and the $L^2$ quotient gauge follows Kernel quotient and gauge for discrete curl-curl, restricted to this PEC subspace. The continuous PEC eigenproblem is to find $\lambda >0$ and $0\mathrm{\ne }u\in X$ such that ${\mu }^{-1}(\mathrm{curl}u,\mathrm{curl}v{)}_{L^2}=\lambda (u,v{)}_{ϵ}(v\in X)\mathrm{.}$ The discrete problem replaces $X$ by $X_h$ and tests against every $v_h\in X_h$. The unquotiented PEC pencil on $V$ is to find $\lambda \ge 0$ and $0\mathrm{\ne }u\in V$ such that the same weak equation holds for every $v\in V$. The positive constants are homogeneous scalar coefficients. The coefficient extensions to variable or tensor-valued media require their own weighted compactness hypotheses and are not inferred from the unweighted PE09 discrete-compactness interface.