Discrete de Rham cohomology and contractible-mesh exactness

If $\mathcal{T}=\mathrm{\varnothing }$, [F12] gives $\mathrm{\Omega }=\mathrm{\varnothing }$; by [F1] and [F4] all real chain, cochain, and finite-element spaces are zero, and [F11] gives $H_k(\mathrm{\varnothing };\mathbb{R})=0$. All cohomology and harmonic statements then reduce to the unique maps and decompositions of zero spaces, and the nonempty contractible branch is inapplicable. Suppose henceforth that $\mathcal{T}\mathrm{\ne }\mathrm{\varnothing }$. In degree zero, the vertex basis ${\lambda }_i$ is dual to vertex evaluation; a value at each vertex determines a unique affine function on each incident tetrahedron, and common-face traces agree because they agree at the face vertices. Thus $J^0:W_h^0\to C^0$ is a bijection. In degree one, [F2] makes the six local edge moments a bijection between each local Nedelec space and its six edge values. The definition of $N_h^1$ imposes one value for each globally oriented edge. Any global assignment therefore determines the unique local fields on each cell, and [F3] makes their assembly a global H(curl) field; conversely, zero global moments force every cell restriction to vanish by [F2]. Thus $J^1:N_h^1\to C^1$ is a bijection.

In degree two, the four oriented face fluxes determine each local $R{T}_0$ field by [F3]. The definition of $N_h^2$ imposes one value for every global oriented face; these values determine unique local fields and [F3] assembles them into a global H(div) field. Zero global fluxes force every local field to vanish, so $J^2:N_h^2\to C^2$ is a bijection. In degree three, a constant is determined by its cell integral, so $J^3:N_h^3\to C^3$ is a bijection.

The dual differential ${\delta }^k$ evaluates a cochain on the oriented boundary of each $(k+1)$-simplex. For $k=0$, its value on an oriented edge is the terminal nodal value minus the initial value; for $k=1$, its value on a face is the signed sum of its boundary-edge circulations; and for $k=2$, its value on a tetrahedron is the signed sum of its boundary-face fluxes. By [F3] these are exactly the degrees of freedom of $\mathrm{\nabla }$, $\mathrm{curl}$, and $\mathrm{div}$. Thus $J^{k+1}{d}_h={\delta }^k{J}^k$ for each differential. By [F8], ${\delta }^{k+1}{\delta }^k=0$, so these maps are cochain complexes and the bijections induce isomorphisms on cohomology.

Write $Z_k=\ker {\mathrm{\partial }}_k$ and $B_k=\mathrm{im}{\mathrm{\partial }}_{k+1}$; [F8] gives $B_k\subseteq Z_k$. Identify each cochain with its coordinate functional on the oriented simplex basis. Then $\ker {\delta }^k$ consists exactly of functionals vanishing on $B_k$, namely $B_k^{\perp }$. The image of ${\delta }^{k-1}={\mathrm{\partial }}_k^{\ast }$ lies in $Z_k^{\perp }$, since $({\mathrm{\partial }}_k^{\ast }\mathrm{\ell })(z)=\mathrm{\ell }({\mathrm{\partial }}_{k}z)=0$ for $z\in Z_k$. These two subspaces have the same dimension: $\dim \mathrm{im}{\mathrm{\partial }}_k^{\ast }=\mathrm{rank}{\mathrm{\partial }}_k=\dim C_k-\dim Z_k=\dim Z_k^{\perp }$, so $\mathrm{im}{\mathrm{\partial }}_k^{\ast }=Z_k^{\perp }$. Restricting a functional in $B_k^{\perp }$ to $Z_k$ therefore descends to $Z_k\mathrm{/}{B}_k$ and has kernel $Z_k^{\perp }$. Every functional on $Z_k\mathrm{/}{B}_k$ pulls back to a functional on $Z_k$ vanishing on $B_k$; extend a finite basis of $Z_k$ to one of $C_k$ and set the added basis values to zero. This proves surjectivity of restriction and hence $H^k(C^{\ast })=B_k^{\perp }\mathrm{/}{Z}_k^{\perp }\cong {\mathrm{Hom}}_{\mathbb{R}}(Z_k\mathrm{/}{B}_k,\mathbb{R})={\mathrm{Hom}}_{\mathbb{R}}(H_k^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(\mathcal{T};\mathbb{R}),\mathbb{R})$. Since these spaces are finite dimensional, their real duals have the same dimension.

By [F5], $H_k^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(\mathcal{T};\mathbb{R})\cong H_k(\mathrm{\mid }\mathcal{T}\mathrm{\mid };\mathbb{R})$. If $\mathrm{\mid }\mathcal{T}\mathrm{\mid }=\bar{\mathrm{\Omega }}$ is nonempty, connected, and contractible in the sense of [F7], [F6] makes this group $\mathbb{R}$ for $k=0$ and zero for $k>0$. Steps 1.1–2.2 therefore show that the discrete cohomology is one-dimensional in degree zero and zero in positive degrees, which is exactness of the displayed finite sequence, including surjectivity of its last map. The globally constant functions lie in $W_h^0$ and have zero gradient, so the one-dimensional degree-zero kernel consists exactly of constants.

For arbitrary topology, let $Z_h^1=\ker \mathrm{curl}$ and $B_h^1=\mathrm{\nabla }{W}_h^0$. By step 3.1, $B_h^1\subseteq Z_h^1$. The space $Z_h^1$ is finite dimensional as a subspace of $N_h^1$, so [F10] applied inside $Z_h^1$ gives $Z_h^1=B_h^1\oplus (Z_h^1\cap (B_h^1{)}^{\perp })$ with orthogonality from [F9]. The second summand has dimension $\dim (Z_h^1\mathrm{/}{B}_h^1)=\dim H^1(C^{\ast })=\dim H_1(\mathrm{\mid }\mathcal{T}\mathrm{\mid };\mathbb{R})$ by steps 3.1–2.2 and the duality proved in step 4.1. Applying the same [F10] argument to $Z_h^2=\ker \mathrm{div}$ and $B_h^2=\mathrm{curl}{N}_h^1\subseteq Z_h^2$ gives the asserted second harmonic space and dimension. These are discrete cohomology components, not an assertion that a continuum sequence is exact.