Shape-regular affine P1 interpolation estimates for C2 functions

Fix a cell $K$ and a vertex $a_0$. Taylor's formula along the segment from $a_0$ to $x\in K$ gives $u(x)=p_K(x)+r_K(x)$ with $p_K(x)=u(a_0)+\mathrm{\nabla }u(a_0)\cdot (x-a_0)$, $\mathrm{\mid }{r}_K(x)\mathrm{\mid }\le \frac{1}{2}{h}_K^2\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$, and $\mathrm{\mid }\mathrm{\nabla }{r}_K(x)\mathrm{\mid }\le h_K\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$.

The nodal interpolant reproduces $p_K$, so $u-I_{K}u=r_K-{\sum }_i{r}_K(a_i){\lambda }_i$. On $K$, $0\le {\lambda }_i\le 1$ and ${\sum }_i{\lambda }_i=1$. The altitude from any vertex to its opposite facet is at least ${\rho }_K$, hence $\mathrm{\mid }\mathrm{\nabla }{\lambda }_i\mathrm{\mid }\le {\rho }_K^{-1}\le \sigma h_K^{-1}$ by [F1]. Substitution of the bounds in step 1.1 gives $\mathrm{\mid }u-I_{K}u\mathrm{\mid }\le C(d)h_K^2\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$ and $\mathrm{\mid }\mathrm{\nabla }(u-I_{K}u)\mathrm{\mid }\le C(d,\sigma )h_K\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$.

Squaring the two pointwise bounds, integrating on each cell, summing over the finite mesh, using $h_K\le h$ and ${\sum }_K\mathrm{\mid }K\mathrm{\mid }=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$, and taking square roots proves both global estimates with $M_2=\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{\mathrm{\infty }}$.

On a shared interior facet, the affine traces from its two incident cells agree at all facet vertices, so they agree everywhere on that facet; thus the cell interpolants assemble to a continuous $I_{h}u$. If $u$ vanishes on a boundary facet, all its vertices there have zero nodal values, so the affine trace also vanishes there. This proves both global conformity and the boundary claim.