Whitney edge, face, and cell moments commute with grad, curl, and div

For each cell $K$, define $I_K^{0}f$ by its four vertex values, $I_K^{1}u$ by its six edge circulations, $I_K^{2}v$ by its four face-normal fluxes, and $I_K^{3}g$ by its cell integral. The vertex basis ${\lambda }_i$ is dual to vertex evaluation. By [F1], the local edge-moment map is bijective, so any six prescribed edge values determine one local field; in particular they define $I_K^{1}u$. For $R{T}_0(K)$, the face flux map is injective: if all four fluxes of $q(x)=a+cx$ vanish, [F3] gives $3c\mathrm{\mid }K\mathrm{\mid }={\int }_K\mathrm{div}q=0$, so $c=0$; then $a\cdot n_F=0$ for all four face normals. Three reference-tetrahedron face area vectors are linearly independent, and the invertible cofactor map in [F5] preserves this independence on $K$, so $a=0$. Since $R{T}_0(K)$ has dimension four, the four moments are bijective and any four prescribed values determine one field, defining $I_K^{2}v$. A constant is determined by its cell integral, so $I_K^{3}g$ is also unique.

On a shared triangular face, the tangential trace of a local Nedelec field is a linear combination of the three Whitney edge traces on that face: a basis function from an edge containing the opposite vertex has zero tangential trace because the gradient of the opposite barycentric coordinate is normal to the face. The three face-edge circulations are unisolvent by the same endpoint calculation as [F1]. A global assignment gives the same oriented circulation from either incident tetrahedron, so the two tangential traces agree. Cellwise curl integration by parts, using [F3] on $u\times \phi$ for a compactly supported smooth test $\phi$, then cancels all interior face terms by [F4]; boundary terms vanish because $\phi$ is compactly supported. Hence the assembled edge field has a distributional curl equal to its piecewise polynomial curl in $L^2$ and lies in the global $H(\mathrm{curl})$ space.

The identity ${\sum }_{j\mathrm{\ne }i}{\varphi }_{ij}=-\mathrm{\nabla }{\lambda }_i$ (where ${\varphi }_{ji}:=-{\varphi }_{ij}$) follows from ${\sum }_j{\lambda }_j=1$ and ${\sum }_j\mathrm{\nabla }{\lambda }_j=0$; hence $\mathrm{\nabla }{I}_K^{0}f\in N_0(K)$. On an oriented edge $e=[x_i,x_j]$, [F6] gives ${\int }_e\mathrm{\nabla }{I}_K^{0}f\cdot t_{e}ds=I_K^{0}f(x_j)-I_K^{0}f(x_i)=f(x_j)-f(x_i)={\int }_e\mathrm{\nabla }f\cdot t_{e}ds$. The two fields lie in the local Nedelec space, and [F1] makes the edge moments unisolvent, so $\mathrm{\nabla }{I}_K^{0}f=I_K^1\mathrm{\nabla }f$ on every cell.

Each $\mathrm{curl}{\varphi }_{ij}=2\mathrm{\nabla }{\lambda }_i\times \mathrm{\nabla }{\lambda }_j$ is constant, so $\mathrm{curl}{I}_K^{1}u\in R{T}_0(K)$. For every oriented face $F$, Stokes [F2] gives ${\int }_F\mathrm{curl}(I_K^{1}u)\cdot n_{F}dS={\sum }_{e\subset \mathrm{\partial }F}\mathrm{sgn}(e,F){\int }_e{I}_K^{1}u\cdot t_{e}ds={\int }_F\mathrm{curl}u\cdot n_{F}dS$. The first equality uses the affine moment convention [F5]; the second uses the defining edge moments. Both curls are in $R{T}_0(K)$ and the four face moments are unisolvent by step 1.1, so $\mathrm{curl}{I}_K^{1}u=I_K^2\mathrm{curl}u$.

On a tetrahedron $K$, Gauss–Green [F3] gives ${\int }_K\mathrm{div}(I_K^{2}v)dx={\sum }_{F\subset \mathrm{\partial }K}\mathrm{sgn}(F,K){\int }_F{I}_K^{2}v\cdot n_{F}dS={\int }_K\mathrm{div}vdx$. Since $\mathrm{div}(a+cx)=3c$ is constant, equality of these cell integrals proves $\mathrm{div}{I}_K^{2}v=I_K^3\mathrm{div}v$ on $K$. More generally, every local RT0 normal trace is constant on a planar face because $x\cdot n_F$ is constant there; a shared oriented face moment therefore determines the same physical normal component from both cells. Summing cellwise Gauss–Green against a compact test function cancels interior terms by [F4], while boundary terms vanish, so the assembled face field has distributional divergence equal to its piecewise constant divergence in $L^2$.

For any discrete $q_h\in W_h^0$, $u_h\in N_h^1$, or $v_h\in N_h^2$, its restriction to each cell is a polynomial and extends smoothly to a neighborhood of that cell. Apply the local identities in steps 1.3, 2.1 and 2.2 to these restrictions. The local interpolant is the identity on the corresponding finite-element input by the local moment unisolvence and construction in step 1.1, so the identities show $\mathrm{\nabla }{q}_h{\mathrm{\mid }}_K$, $\mathrm{curl}{u}_h{\mathrm{\mid }}_K$, and $\mathrm{div}{v}_h{\mathrm{\mid }}_K$ lie in their next local spaces. The edge moments of $\mathrm{\nabla }{q}_h$ are endpoint differences of shared vertex values by step 1.3. The face fluxes of $\mathrm{curl}{u}_h$ are signed sums of the shared global edge values by step 2.1, so adjacent cells assign the same oriented flux to every shared face. The divergence target $N_h^3$ has only cellwise constant data and imposes no cross-cell matching. The assemblies in steps 1.2 and 2.2 give the required H(curl) and H(div) membership. Hence all three discrete differentials map between the stated global spaces, proving the subcomplex assertion.