Backward-Euler incremental pressure correction on a periodic staggered grid

Fix $d\ge 1$, $N\ge 2$, $h=1\mathrm{/}N$, viscosity $\nu >0$, and time step $k>0$. Given $u^n\in V_h$ with $D_h{u}^n=0$, a mean-zero pressure $p^n\in Q_h$, and a force $f^{n+1}\in V_h$ whose components each have zero discrete mean, $h^d{\sum }_{j\in J_N}{f}_{i,j}^{n+1}=0(1\le i\le d),$ define the next step by the following two solves and update:

1. Find ${\tilde{u}}^{n+1}\in V_h$ satisfying $\frac{{\tilde{u}}^{n+1}-u^n}{k}+\nu A_h{\tilde{u}}^{n+1}+G_h{p}^n=f^{n+1}\mathrm{.}$

2. Find the unique ${\varphi }^{n+1}\in Q_h$ satisfying ${\mathrm{\Delta }}_h{\varphi }^{n+1}=k^{-1}{D}_h{\tilde{u}}^{n+1}\mathrm{.}$

3. Set $u^{n+1}={\tilde{u}}^{n+1}-k{G}_h{\varphi }^{n+1},p^{n+1}=p^n+{\varphi }^{n+1}\mathrm{.}$

The pressure is kept in $Q_h$ after every solve; the constant pressure mode is set to zero. The procedure is only asserted for the periodic staggered grid and the admissible data stated here. The separate well-definedness theorem Well-definedness of the periodic pressure-correction solves ↗ verifies both solves and the preserved constraints.