Finite-volume companions, mean tests and weak-gradient closure

Write $j_{K,F}=v_F-v_K$. The moment cancellation gives $G_{K}v=\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\sum }_F\mathrm{\mid }F\mathrm{\mid }{j}_{K,F}{n}_{K,F}$. Hölder, the bounded face count and [F1] imply $\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }{G}_{K}v{\mathrm{\mid }}^p\le C{\sum }_F{h}^{d-p}\mathrm{\mid }{j}_{K,F}{\mathrm{\mid }}^p$. Since ${\rho }_{K,F}=j_{K,F}-G_{K}v\cdot (x_F-x_K)$ and $\mathrm{\mid }{x}_F-x_K\mathrm{\mid }\le Ch$, summing gives $B_{h,p}\le C{D}_{h,p}$. Conversely $j_{K,F}={\rho }_{K,F}+G_{K}v\cdot (x_F-x_K)$ gives $D_{h,p}\le C{B}_{h,p}$.

If $K,L$ share $F$, $v_K-v_L=j_{L,F}-j_{K,F}$. Along a supplied vertex-star path, the difference between $v_K$ and any other incident cell value is a sum of at most $2J$ such increments. At a boundary vertex the same telescoping ends at a zero boundary face. Hence for every simplex $S\subset \bar{K}$ and its vertex $z$, $\mathrm{\mid }(E_{h}v)(z)-v_K{\mathrm{\mid }}^p\le C{\sum }_{(L,F)\in \mathcal{N}(K)}\mathrm{\mid }{j}_{L,F}{\mathrm{\mid }}^p$, where $\mathcal{N}(K)$ is the union of the finitely many supplied paths at its vertices. These sets have bounded cardinality and bounded overlap: a cell has at most $J(d+1)$ subdivision vertices and each vertex has at most $J$ incident cells.

The scaled simplex estimates [F2] first bound the difference between the mean on a simplex and the mean on one of its facets by $C{h}^{1-d\mathrm{/}p}\mathrm{\parallel }\mathrm{\nabla }z{\mathrm{\parallel }}_{L^p(S)}$. Means on adjacent simplices differ by the sum of their differences to the shared facet mean, since their traces agree. The fan subdivision is a connected finite union of at most $J$ simplices. Telescoping between their means, and averaging over each original cell or original face, therefore gives $\mathrm{\mid }(I_{h}z{)}_F-(I_{h}z{)}_K{\mathrm{\mid }}^p\le C{h}^{p-d}{\int }_K\mathrm{\mid }\mathrm{\nabla }z{\mathrm{\mid }}^p$. The same trace argument applies to boundary faces; zero-boundary approximation makes their means zero. Multiplication by $\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}^{1-p}\asymp h^{d-p}$ and summation proves $D_{h,p}(I_{h}z)\le C\mathrm{\parallel }\mathrm{\nabla }z{\mathrm{\parallel }}_p$.

For smooth $z$, Gauss–Green gives $G_K{I}_{h}z=\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_K\mathrm{\nabla }z$ exactly. On each cell Taylor expansion around its centroid and the corresponding face averages show $\mathrm{\mid }(I_{h}z{)}_F-(I_{h}z{)}_K-G_K{I}_{h}z\cdot (x_F-x_K)\mathrm{\mid }\le C{h}^2\mathrm{\parallel }{D}^{2}z{\mathrm{\parallel }}_{\mathrm{\infty }}$ under the stated centroid hypothesis. Thus for this mean interpolator the residual sum is at most $C{h}^p\mathrm{\parallel }{D}^{2}z{\mathrm{\parallel }}_{\mathrm{\infty }}^p$, because there are $O(h^{-d})$ cells. Uniform continuity of $\mathrm{\nabla }z$ gives the asserted gradient consistency.

Barycentric coordinates on $S$ have absolute value at most one and gradients at most $C\mathrm{/}h$, by the inradius bound. Applying step 1.2 to $E_{h}v-v_K$ gives its $p$th-power $L^p(S)$ norm at most $C{h}^d$ times the path sum, and its gradient norm at most $C{h}^{d-p}$ times that sum. Summing with bounded overlap proves $\mathrm{\parallel }{E}_{h}v-{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_p\le Ch{D}_{h,p}$ and $\mathrm{\parallel }\mathrm{\nabla }{E}_{h}v{\mathrm{\parallel }}_p\le C{D}_{h,p}$. The continuous piecewise affine function vanishes on all boundary facets. Its zero extension is globally Lipschitz and has compact support: along a segment, the finite simplex pieces and the exterior zero pieces have derivative bounded by the largest simplex gradient and agree at crossings. To verify the zero-boundary closure directly, multiply by $\max (0,\min (1,k\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })-1))$. The error and its gradient are supported in shrinking boundary layers, and $\mathrm{\mid }{E}_{h}v(x)\mathrm{\mid }\le L\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })$ bounds the cutoff-gradient term by $2L$ on the layer $1\mathrm{/}k<\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })<2\mathrm{/}k$. They tend to zero in $W^{1,p}$ by absolute continuity of the integral. Each cutoff is Lipschitz with compact support in $\mathrm{\Omega }$ and can be mollified inside $\mathrm{\Omega }$, producing smooth compactly supported approximants. Thus $E_{h}v\in W_0^{1,p}$, and [F4] proves the full $W^{1,p}$ bound and the $L^p$ state bound.

For $p<2$, the same barycentric estimate in $L^2(S)$ and ${\mathrm{\ell }}^2\le {\mathrm{\ell }}^p$ for the combined finite coefficient list give $\mathrm{\parallel }{E}_{h}v-{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_2\le C{h}^{d\mathrm{/}2}({\sum }_{K,F}\mathrm{\mid }{j}_{K,F}{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}\le C{h}^{1+d\mathrm{/}2-d\mathrm{/}p}{D}_{h,p}$. For $p\ge 2$, finite-measure Hölder and step 2.1 give $Ch{D}_{h,p}$ instead. The exponent is positive precisely on the stated strict branch. The continuous embedding [F4] bounds $\mathrm{\parallel }{E}_{h}v{\mathrm{\parallel }}_2$, proving the state $L^2$ estimate as well.

Let ${\varphi }_K$ and ${\varphi }_F$ denote vector means over cells and faces. Gauss–Green, shared face cancellation and $v_F=0$ on the boundary give exactly $\int G_{h}v\cdot \varphi +{\mathrm{\Pi }}_{h}v\mathrm{div}\varphi ={\sum }_{K,F}\mathrm{\mid }F\mathrm{\mid }{j}_{K,F}{n}_{K,F}\cdot ({\varphi }_K-{\varphi }_F)$. Since $\mathrm{\mid }{\varphi }_K-{\varphi }_F\mathrm{\mid }\le Ch\mathrm{\parallel }\mathrm{\nabla }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}$, Hölder bounds this by $Ch{D}_{h,p}({\sum }_{K,F}\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}{)}^{1\mathrm{/}{p}^{\mathrm{\prime }}}\mathrm{\parallel }\mathrm{\nabla }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}$. The last sum is uniformly bounded by [F1], proving the estimate. Strong and weak convergence pass its left side to $\int g\cdot \varphi +v\mathrm{div}\varphi$, which is zero; interior compactly supported tests identify $g=\mathrm{\nabla }v$. Hölder in time gives the same assertions for space-time bounds.