Primal H¹ approximation on shape-regular affine finite-element meshes

On the reference cell write $\widehat{I}w={\sum }_{i=1}^N{\widehat{\sigma }}_i(w){\widehat{\varphi }}_i$. Each degree of freedom is bounded on $W^{r+1,2}$ by Given, and each fixed polynomial shape function has finite $W^{1,2}$ norm. Thus $\mathrm{\parallel }\widehat{I}w{\mathrm{\parallel }}_{W^{1,2}}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{r+1,2}}$. By [F1], $\widehat{I}q=q$ for every $q\in {\mathbb{P}}_r$. For such $q$, $w-\widehat{I}w=(w-q)-\widehat{I}(w-q),$ so $\mathrm{\parallel }w-\widehat{I}w{\mathrm{\parallel }}_{W^{m,2}}\le \mathrm{\parallel }w-q{\mathrm{\parallel }}_{W^{m,2}}+C\mathrm{\parallel }w-q{\mathrm{\parallel }}_{W^{r+1,2}}(m=0,1)\mathrm{.}$

Choose the polynomial $q\in {\mathbb{P}}_r$ supplied by [F2]. Since $m\le 1\le r+1$, [F2] bounds both terms in step 1.1 by $C\mathrm{\mid }w{\mathrm{\mid }}_{W^{r+1,2}}$. Hence the reference-cell interpolation error obeys $\mathrm{\parallel }w-\widehat{I}w{\mathrm{\parallel }}_{W^{m,2}}\le C\mathrm{\mid }w{\mathrm{\mid }}_{W^{r+1,2}}$ for $m=0,1$.

For a cell $K$ with affine matrix $A_K$, apply [F3] to transform the reference error and the order-$(r+1)$ derivative seminorm back to $K$. The determinant factors cancel; the remaining factor is bounded by $C\mathrm{\parallel }{A}_K^{-1}{\mathrm{\parallel }}^m\mathrm{\parallel }{A}_K{\mathrm{\parallel }}^{r+1}\le C{h}_K^{r+1-m}$ by [F4]. Therefore $\mathrm{\parallel }u-I_{K}u{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{r+1}\mathrm{\mid }u{\mathrm{\mid }}_{W^{r+1,2}(K)},\mathrm{\mid }u-I_{K}u{\mathrm{\mid }}_{W^{1,2}(K)}\le C{h}_K^r\mathrm{\mid }u{\mathrm{\mid }}_{W^{r+1,2}(K)}\mathrm{.}$

Let $E_h:={\bigcup }_{K\in {\mathcal{T}}_h}\mathrm{\partial }K$, a finite union of simplex facets. Each facet lies in a bounded affine hyperplane; choose a coordinate with nonzero normal component and write that hyperplane as the graph of an affine function over a bounded box in ${\mathbb{R}}^{d-1}$. Partition the base box into at most $C_0{N}^{d-1}$ boxes of side $1\mathrm{/}N$. The affine function varies by at most $C_1\mathrm{/}N$ on each, so each graph piece lies in a slightly enlarged half-open $d$-box of volume at most $C_2{N}^{-d}$. These elementary-set cover members have total volume at most $C_0{C}_2\mathrm{/}N$; the outer-measure definition and box-volume formula give outer measure zero for the facet, and completeness makes it Lebesgue-null. Thus $E_h$ is null, and $\mathrm{\Omega }\setminus E_h$ is the disjoint union of the cell interiors by the mesh definition. Square and sum the estimates in step 3.1 over these interiors; the sums of squared Sobolev seminorms equal the corresponding global seminorms. Since $h_K\le h\le 1$, the order-zero contribution $h_K^{r+1}$ is at most $h^r$. The global interpolation map in Given takes values in $V_h$, yielding $\mathrm{\parallel }u-I_{h}u{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}\le C{h}^r\mathrm{\mid }u{\mathrm{\mid }}_{W^{r+1,2}(\mathrm{\Omega })}$.