Local approximation and reproduction of the P1 moment interpolant

The definition gives finitely many coefficients, each an integral against a bounded affine polynomial, hence a linear functional on $L^2(K_z)$; the resulting finite sum is continuous piecewise affine. Its boundary nodal coefficients are zero, so [F2] places it in $V(\mathcal{T})$. If $v\in V(\mathcal{T})$, then on the selected cell $K_z$ its affine restriction is a linear combination of the three local hats. The dual moment identities recover its nodal coefficient at $z$, so $c_z(v)=v(z)$ at every interior vertex; at boundary vertices both the coefficient and the P1 trace value are zero. Therefore $I_{\mathcal{T}}v=v$.

For a cell $K$ not meeting the boundary, set $c=v_{{\omega }_K}$; for a boundary cell set $c=0$. The patch Poincaré estimate gives $\mathrm{\parallel }v-c{\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. For each interior vertex $z$ of $K$, put $a_z=c_z(v)-c={\int }_{K_z}(v-c){\psi }_{z,K_z}$; this equality follows from the dual identities, which give ${\int }_{K_z}{\psi }_{z,K_z}=1$. These are local integrals of an $L^2(K_z)$ function, with no application of $I_{\mathcal{T}}$ to $v-c$. The scaled dual bound gives $\mathrm{\mid }{a}_z\mathrm{\mid }\le C{h}_K^{-1}\mathrm{\parallel }v-c{\mathrm{\parallel }}_{L^2(K_z)}$, since $K_z\subset {\omega }_K$ and its diameter is comparable to $h_K$. At a boundary vertex of $K$, necessarily $c=0$ and the corresponding coefficient difference is zero. The three barycentric hats on $K$ sum to one, so $(I_{\mathcal{T}}v-c){\mathrm{\mid }}_K={\sum }_{z\in \mathcal{V}(K),z\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{a}_z{\phi }_z{\mathrm{\mid }}_K$. Each hat has $L^2(K)$ norm at most $C{h}_K$ and gradient norm at most $C$. Thus $\mathrm{\parallel }{I}_{\mathcal{T}}v-c{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }v-c{\mathrm{\parallel }}_{L^2({\omega }_K)}$ and $\mathrm{\parallel }\mathrm{\nabla }{I}_{\mathcal{T}}v{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{-1}\mathrm{\parallel }v-c{\mathrm{\parallel }}_{L^2({\omega }_K)}$. The patch inequality and $v-I_{\mathcal{T}}v=(v-c)-(I_{\mathcal{T}}v-c)$ prove the first two bounds. This argument uses only that each selected $K_z$ is incident to $z$, so its constants are uniform over all such selections.

Apply the scaled trace inequality to $v-I_{\mathcal{T}}v$ on $K$. The first two bounds from step 1.2 control the cell $L^2$ term and its gradient; the neighboring patch gives the same estimates on $K^{\mathrm{\prime }}$. Since $h_E$ and $h_K$ are comparable by shape regularity, this yields the asserted edge bound.