Periodic staggered-grid velocity and pressure spaces

Fix an integer dimension $d\ge 1$ and an integer resolution $N\ge 2$, where $N\in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)), and put $h=1\mathrm{/}N$. Real scalar and array operations use the field $\mathbb{R}$ (The reals form a field). Let $J_N=\{0,\dots ,N-1{\}}^d$; each coordinate index addition below is reduced to the unique element of $\{0,\dots ,N-1\}$ congruent modulo $N$. The unit torus ${\mathbb{T}}^d$ means the unit cube with opposite faces identified, so fields and arrays are periodic across those faces. Write $\mathbf{1}=(1,\dots ,1)$ and let $e_i$ be the $i$th standard coordinate vector. Cell $j$ has centre $x_j=h(j+\frac{1}{2}\mathbf{1})$. The pressure value $q_j$ is stored at that centre. Velocity component $i$ is stored at the right face of cell $j$, with centre $x_j+\frac{h}{2}{e}_i$.

Let ${\tilde{Q}}_h$ be all real scalar cell arrays and let $Q_h=\{q\in {\tilde{Q}}_h:h^d{\sum }_j{q}_j=0\}$. Let $V_h$ be the real vector face arrays $v=(v_{i,j}{)}_{1\le i\le d,j\in J_N}$. Give them the mass inner products $(q,r{)}_{Q_h}=h^d{\sum }_{j\in J_N}{q}_j{r}_j,(v,w{)}_{M_h}=h^d{\sum }_{i=1}^d{\sum }_{j\in J_N}{v}_{i,j}{w}_{i,j}\mathrm{.}$ Define the pressure gradient $G_h:{\tilde{Q}}_h\to V_h$ and velocity divergence $D_h:V_h\to Q_h$ by $(G_{h}q{)}_{i,j}=\frac{q_{j+e_i}-q_j}{h},(D_{h}v{)}_j={\sum }_{i=1}^d\frac{v_{i,j}-v_{i,j-e_i}}{h}\mathrm{.}$ The compatible pressure Laplacian is ${\mathrm{\Delta }}_h=D_h{G}_h$; its restriction to $Q_h$ is negative semidefinite, and $G_h$ annihilates constants. Define the componentwise vector Laplacian by $({\mathrm{\Delta }}_h^{V}v{)}_{i,j}=h^{-2}{\sum }_{\mathrm{\ell }=1}^d(v_{i,j+e_{\mathrm{\ell }}}-2v_{i,j}+v_{i,j-e_{\mathrm{\ell }}}),A_h=-{\mathrm{\Delta }}_h^V,$ and the discrete velocity-gradient seminorm by $\mathrm{\parallel }{\mathcal{G}}_{h}v{\mathrm{\parallel }}_{M_h}^2=h^{d-2}{\sum }_{i=1}^d{\sum }_{\mathrm{\ell }=1}^d{\sum }_{j\in J_N}\mathrm{\mid }{v}_{i,j+e_{\mathrm{\ell }}}-v_{i,j}{\mathrm{\mid }}^2\mathrm{.}$ The fixed cell/face placement is part of the definition; collocated central differences are a different discretization.