Discrete Helmholtz projection and pressure Schur identities

The mass matrix $M$ is symmetric positive definite because for a nonzero coefficient vector $x$, $x^{T}Mx=\mathrm{\parallel }{v}_h(x){\mathrm{\parallel }}_{L^2}^2>0$. The inf-sup condition in [F1] says that $B^{T}y=0$ for a mean-zero pressure coefficient vector $y$ implies $y=0$; hence $B$ has full row rank on the gauged pressure coordinates.

Therefore $S_M:=B{M}^{-1}{B}^T$ is symmetric positive definite: for $y\mathrm{\ne }0$, $y^T{S}_{M}y=(B^{T}y{)}^T{M}^{-1}(B^{T}y)>0$. The formula for $P_M$ is defined. Direct multiplication gives $B{P}_M=B-B{M}^{-1}{B}^T{S}_M^{-1}B=0$, and if $Bx=0$ then $P_{M}x=x$. Thus its range is exactly $\ker B$.

For every $x$, $x-P_{M}x=M^{-1}{B}^T{S}_M^{-1}Bx$. If $z\in \ker B$, then $z^{T}M(x-P_{M}x)=(Bz{)}^T{S}_M^{-1}Bx=0$. Hence $x-P_{M}x$ is mass-orthogonal to the kernel and $P_M$ is the mass-orthogonal projector.

By [F3], $A$ is symmetric positive definite. Full row rank of $B$ from step 2.1 then makes $S_A:=B{A}^{-1}{B}^T$ positive definite by the same quadratic-form argument as in step 2.1. Eliminating velocity from $Au+B^{T}p=f$, $Bu=g$ gives $S_{A}p=B{A}^{-1}f-g$, which is the pressure Schur identity.

If pressure coordinates are instead taken in the full continuous P1 space, [F2] puts constants in $\ker B^T$. Conversely, if $B^{T}y=0$, the corresponding $q_h\in S_h^1$ pairs to zero with every velocity. Its mean-zero part $q_h-\mathrm{\mid }D{\mathrm{\mid }}^{-1}{\int }_D{q}_h$ also pairs to zero by [F2]; the stable inf-sup in [F1] forces that part to vanish. Thus the nullspace consists exactly of constants, and restricting to $Q_h=L_0^2\cap S_h^1$ removes that gauge mode.