Absolute and PEC-relative discrete Maxwell zero modes

Let T be a finite matching affine tetrahedral mesh whose realization is closure(Omega), with Omega a nonempty bounded Lipschitz polyhedron. Let W_h^0,N_h^1,N_h^2,N_h^3 be the full lowest-order Whitney de Rham spaces of Lowest-order Whitney and Nedelec de Rham spaces, and use their real L^2 inner products. Define

${\mathcal{H}}_h^1:=\ker (\mathrm{curl}{\mathrm{\mid }}_{N_h^1})\cap (\mathrm{\nabla }{W}_h^0{)}^{\perp },{\mathcal{H}}_h^2:=\ker (\mathrm{div}{\mathrm{\mid }}_{N_h^2})\cap (\mathrm{curl}{N}_h^1{)}^{\perp },{\beta }_k:={\dim }_{\mathbb{R}}{H}_k(\mathrm{\mid }\mathcal{T}\mathrm{\mid };\mathbb{R})(k=1,2)\mathrm{.}$

Under Countable Choice, dim_R H_h^1=beta_1 and dim_R H_h^2=beta_2 for the full complex. These are the harmonic complements for the full, absolute finite-element complex.

For the separate PEC-relative family, assume the full Axiom of Choice, including its Countable Choice and Dependent Choice consequences, and let {T_h} be a shape-regular family of matching affine tetrahedral meshes of the same Omega with h=max_K diam(K)->0. Assume also the exact full-Dirichlet H(curl) regular decomposition and the relative commuting-projection hypotheses stated in Facts & Assumptions below. For each mesh write V_h:=N_h^1 intersect H_0(curl;Omega), S_h:=W_h^0 intersect H_0^1(Omega), K_h:=ker(curl|_{V_h}), B_h:=grad S_h, and H_h^rel:=K_h intersect B_h^\perp, with orthogonality in real L^2. Put

$X_h:=\{v_h\in V_h:(v_h,\mathrm{\nabla }{q}_h{)}_{L^2}=0\text{ for every }{q}_h\in S_h\},Z_h:=V_h\cap K_h^{\perp }\mathrm{.}$

Then K_h=B_h direct-sum H_h^rel, every sequence v_h in X_h bounded in H(curl;Omega) has an L^2-convergent subsequence, and there is a constant C independent of h such that ||v_h||_2<=C||curl v_h||_2 for every v_h in Z_h; the constant C has units metres when v_h is an electric field. If H_h^rel contains a nonzero vector for a mesh, that vector lies in X_h and has zero curl, so no curl Poincare inequality holds on that X_h. The full-complex absolute Betti count in the first clause is not asserted for this relative PEC kernel.

PEC Whitney edge-face Maxwell semidiscretization

Discrete de Rham cohomology and contractible-mesh exactness

A bounded regular decomposition with both summands satisfying full Dirichlet conditions

Uniformly bounded commuting projection for lowest-order Nedelec spaces

Discrete compactness and uniform Poincare inequality for Nedelec fields

The diagram gives the moment-coordinate identification for the full complex. The vertical arrows send a piecewise scalar to its vertex values, an edge field to its oriented edge circulations, a face field to its oriented face fluxes, and a cellwise constant to its cell integrals. The bottom arrows are oriented simplicial coboundaries. The commuting squares mean that the edge, face, and cell moments of grad, curl, and div are the signed sums on the corresponding oriented boundaries. This is the finite-complex interface proved by the cited mathematical supplier; no proof step below relies on the drawing.

\begin{tikzcd}
W_h^0 \arrow[r,"\nabla"] \arrow[d,"\cong"'] &
N_h^1 \arrow[r,"\operatorname{curl}"] \arrow[d,"\cong"'] &
N_h^2 \arrow[r,"\operatorname{div}"] \arrow[d,"\cong"'] &
N_h^3 \arrow[d,"\cong"] \\
C^0(\mathcal T;\mathbb R) \arrow[r,"\delta^0"] &
C^1(\mathcal T;\mathbb R) \arrow[r,"\delta^1"] &
C^2(\mathcal T;\mathbb R) \arrow[r,"\delta^2"] &
C^3(\mathcal T;\mathbb R)
\end{tikzcd}