Approximation by the MINI solenoidal energy projection

For any $z\in V$, equip $Z_h$ with the inner product $a_{\nu }$. It is positive definite by [F1]. The functional $w_h\mapsto a_{\nu }(z,w_h)$ is bounded on this finite-dimensional Hilbert space, so [F2] gives a unique $S_{h}z\in Z_h$ satisfying the displayed equation. Linearity follows from uniqueness and linearity of the right-hand side. For $z\in Z_h$ the equation gives $S_{h}z=z$, so this extension to $V$ is a projection. For any $w_h\in Z_h$, orthogonality gives $a_{\nu }(z-w_h,z-w_h)=a_{\nu }(z-S_{h}z,z-S_{h}z)+a_{\nu }(S_{h}z-w_h,S_{h}z-w_h)\mathrm{.}$ Using $\nu \mathrm{\parallel }r{\mathrm{\parallel }}_V^2\le a_{\nu }(r,r)\le 2\nu \mathrm{\parallel }r{\mathrm{\parallel }}_V^2$ yields the stated best-approximation bound.

Let $F(v)=a_{\nu }(z,v)+b(v,q)$ for $v\in V$. It is bounded on $V$, and $b(z,r)=0$ for every $r\in Q$ because $z\in V_{\sigma }$; hence $(z,q)$ solves the continuous Stokes problem with data $(F,0)$. Let $w_h=I_{h}z$ and $r_h=I_{h}q-\mathrm{\mid }D{\mathrm{\mid }}^{-1}{\int }_D{I}_{h}q$. By [F3], $w_h\in V_h$, $r_h\in Q_h$, and ${\delta }_u:=\mathrm{\parallel }z-w_h{\mathrm{\parallel }}_V\le Ch\mathrm{\parallel }z{\mathrm{\parallel }}_{H^2}$, ${\delta }_q:=\mathrm{\parallel }q-r_h{\mathrm{\parallel }}_Q\le Ch\mathrm{\parallel }q{\mathrm{\parallel }}_{H^1}$. The MINI mixed problem has a solution $(z_h,q_h)$ with $z_h\in Z_h$. Its inf-sup constant satisfies ${\beta }_h^{-1}\le C$ independently of $h$ and $\nu$, while $M_a=2\nu$, $M_b=\sqrt{d}$, and ${\alpha }_{\ast }=\nu$. The mixed error formula therefore gives $R_h\le C{\delta }_u$ and $U_h={\delta }_u+{\nu }^{-1}\{2\nu ({\delta }_u+R_h)+\sqrt{d}{\delta }_q\}+R_h\le C({\delta }_u+{\nu }^{-1}{\delta }_q)$. Step 1.1 with $w_h=z_h$ now yields $\mathrm{\parallel }z-S_{h}z{\mathrm{\parallel }}_V\le \sqrt{2}{U}_h\le C_0(1+{\nu }^{-1})h(\mathrm{\parallel }z{\mathrm{\parallel }}_{H^2}+\mathrm{\parallel }q{\mathrm{\parallel }}_{H^1})$, with $C_0$ independent of $h$ and $\nu$.