Second-order L2 approximation by the MINI Fortin map

Fix a cell $K$ and first take one scalar component of $v$. Let $p$ be the affine patch approximant from [F1], regarded as a global affine polynomial on $D$. This causes no boundary restriction, because $I_h$ is defined on all of $H^1(D)$. Set $w=v-p$. Affine reproduction gives $v-I_{h}v=w-I_{h}w$ on $K$. Apply the first-order local $L^2$ estimate in [F1] to $w$ and its patch gradient estimate to obtain $\mathrm{\parallel }v-I_{h}v{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }(v-p){\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. Apply this componentwise and sum squares for vector $v$. The local gradient bound $\mathrm{\parallel }\mathrm{\nabla }(v-I_{h}v){\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$ is also [F1]. All these estimates use facet averages and require no representative with point values.

Write $e_K=v-I_{h}v$ on $K$. The actual numerator of the Fortin coefficient is the unweighted integral of $e_K$, so [F3] and [F5] give $\mathrm{\mid }{c}_K\mathrm{\mid }\le \mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}2}({\int }_K{b}_K{)}^{-1}\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\mid }K{\mathrm{\mid }}^{-1\mathrm{/}2}\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}$. Hence $\mathrm{\parallel }{b}_K{c}_K{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}$ and $\mathrm{\parallel }\mathrm{\nabla }(b_K{c}_K){\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{-1}\mathrm{\parallel }{e}_K{\mathrm{\parallel }}_{L^2(K)}$. No bound involving a global inverse mesh scale is used.

Bubble interiors are disjoint, so on $K$ the existing map satisfies $v-{\mathrm{\Pi }}_{h}v=e_K-b_K{c}_K$. The triangle inequality, step 1.2 and step 1.1 imply $\mathrm{\parallel }v-{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$ and $\mathrm{\parallel }\mathrm{\nabla }(v-{\mathrm{\Pi }}_{h}v){\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. The zero-boundary membership of ${\mathrm{\Pi }}_{h}v$ is already ensured by [F2].

Square and sum the two estimates in step 2.1. Since $h_K\le h$ and the patch overlap is uniformly bounded by [F1], ${\sum }_K{h}_K^4\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}^2\le C{h}^4\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_2^2$ and ${\sum }_K{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}^2\le C{h}^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_2^2$. Taking square roots proves the two global approximation estimates, with the asserted dependence of constants.

Suppose $v\in H^2(D{)}^d\cap V_{\sigma }$. Because $Q_h\subset L_0^2(D)$, [F4] gives $b(v,q_h)=0$ for every $q_h\in Q_h$. Fortin preservation [F2] therefore puts ${\mathrm{\Pi }}_{h}v\in Z_h$. Best approximation [F4] and step 3.1 give $\mathrm{\parallel }(I-P_{Z_h})v{\mathrm{\parallel }}_2\le \mathrm{\parallel }v-{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_2\le C{h}^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_2$. This uses only $L^2$ orthogonality, with no $H^1$ estimate for $P_{Z_h}$.

Finally let $X=H^2(D{)}^d\cap V_{\sigma }$ with its $H^2$ norm and suppose $u\in W^{1,\mathrm{\infty }}(0,T;X)$. By this Bochner Sobolev hypothesis, $u(t)$ and $u_t(t)$ belong to $X$ for almost every $t$, and their $H^2$ norms have the stated essential bounds. Apply step 4.1 separately to these two fields at such times and take essential suprema. The two claimed time-dependent estimates follow. Together with steps 2.1 and 3.1 this proves every assertion.