The mini-element Fortin map approximates in L2

Put $e_K=v-I_{h}v$. By Holder and [F3], $\mathrm{\mid }{c}_K\mathrm{\mid }\le \frac{\mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}2}}{{\int }_K{b}_K}\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{-d\mathrm{/}2}\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}\mathrm{.}$ Therefore $\mathrm{\parallel }{b}_K{c}_K{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}$.

If $v\in V_{\sigma }$, then $b(v,q_h)=0$ for all $q_h\in Q_h$ by definition of $V_{\sigma }$ and $b$. Preservation [F4] gives $b({\mathrm{\Pi }}_{h}v,q_h)=0$ for every $q_h$, so ${\mathrm{\Pi }}_{h}v\in Z_h$.

Let $v\in C_c^{\mathrm{\infty }}(D{)}^d$ and extend it by zero to ${\mathbb{R}}^d$. On each cell $K$, choose a vertex $x_K$ and let $p_K$ be the affine Taylor polynomial of $v$ there. Every selected facet used in $I_h$ for a vertex of $K$ lies within $C{h}_K$ of $x_K$ by the patch geometry in the face-average interpolation supplier. Apply [F6] on the convex open set ${\mathbb{R}}^d$; the pointwise Taylor error on those facets is at most $C{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{\mathrm{\infty }}$. The scaled dual moments in [F5] then bound the coefficients of $I_h(v-p_K)$ on $K$ by the same amount. Since $I_h$ reproduces $p_K$, this yields $\mathrm{\parallel }v-I_{h}v{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}\le C{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $\mathrm{\mid }{c}_K\mathrm{\mid }\le C{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{\mathrm{\infty }}$. Bubble scaling [F3] gives $\mathrm{\parallel }\mathrm{\nabla }(b_K{c}_K){\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{d\mathrm{/}2+1}\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{\mathrm{\infty }}$. Also $h_K^d\le C\mathrm{\mid }K\mathrm{\mid }$ by [F3], so summing squares gives $\mathrm{\parallel }\mathrm{\nabla }({\mathrm{\Pi }}_{h}v-I_{h}v){\mathrm{\parallel }}_2\le Ch\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{\mathrm{\infty }}$. The H2-to-H1 estimate in [F5] gives $\mathrm{\parallel }{I}_{h}v-v{\mathrm{\parallel }}_V\to 0$, hence ${\mathrm{\Pi }}_{h}v\to v$ in $V$.

On each cell, $v-{\mathrm{\Pi }}_{h}v=e_K-b_K{c}_K$, so step 1.1 and the triangle inequality give $\mathrm{\parallel }v-{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }v-I_{h}v{\mathrm{\parallel }}_{L^2(K)}$. Square, sum over cells, apply [F2], and use bounded patch overlap to obtain $\mathrm{\parallel }v-{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_{L^2(D)}\le Ch\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D)}$.

For general $v\in V$, choose $\varphi \in C_c^{\mathrm{\infty }}(D{)}^d$ with $\mathrm{\parallel }v-\varphi {\mathrm{\parallel }}_V<ϵ$, using [F7]. Uniform Fortin stability [F7] gives $\mathrm{\parallel }{\mathrm{\Pi }}_{h}v-v{\mathrm{\parallel }}_V\le (C_F+1)ϵ+\mathrm{\parallel }{\mathrm{\Pi }}_h\varphi -\varphi {\mathrm{\parallel }}_V$. Let $h\to 0$ by step 1.3 and then $ϵ\downarrow 0$. This proves strong $H_0^1$ convergence for every fixed $v$.