Smooth fields are approximated by the lowest-order edge family

On each cell $K$, [F2] defines the local interpolant $I_{K}w$ by its six edge moments. To verify constants belong to the local space, write ${\varphi }_{ij}={\lambda }_i\mathrm{\nabla }{\lambda }_j-{\lambda }_j\mathrm{\nabla }{\lambda }_i$ with ${\varphi }_{ji}=-{\varphi }_{ij}$. Then ${\sum }_{j\mathrm{\ne }i}{\varphi }_{ij}=-\mathrm{\nabla }{\lambda }_i$ because ${\sum }_j{\lambda }_j=1$. For each coordinate $\mathrm{\ell }$, the barycentric identity $x_{\mathrm{\ell }}={\sum }_i(x_i{)}_{\mathrm{\ell }}{\lambda }_i$ gives $e_{\mathrm{\ell }}=\mathrm{\nabla }{x}_{\mathrm{\ell }}={\sum }_i(x_i{)}_{\mathrm{\ell }}\mathrm{\nabla }{\lambda }_i$, so every constant vector lies in the local span. Unisolvence [F2] then makes $I_K$ reproduce constants. Fix $x_K\in K$ and put $r_K(x)=w(x)-w(x_K)$. Then $\mathrm{\parallel }{r}_K{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}\le h_K\mathrm{\parallel }Dw{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$, so every edge moment of $r_K$ has absolute value at most $h_K^2\mathrm{\parallel }Dw{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$. By [F1], summing the six moment coefficients against their basis functions gives $\mathrm{\parallel }{I}_K{r}_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{5\mathrm{/}2}\mathrm{\parallel }Dw{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$ and $\mathrm{\parallel }{r}_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{5\mathrm{/}2}\mathrm{\parallel }Dw{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$. Consequently $\mathrm{\parallel }{I}_{K}w-w{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{5\mathrm{/}2}\mathrm{\parallel }Dw{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$.

By [F3] the local fields assemble to a global $H(\mathrm{curl})$ field. If $w=0$, then $I_{h}w=0\in H_0(\mathrm{curl};\mathrm{\Omega })$. Otherwise let $d=\mathrm{dist}(\mathrm{supp}w,\mathrm{\partial }\mathrm{\Omega })>0$ and take $h<d\mathrm{/}2$. Any cell with a nonzero edge moment lies within distance $h$ of $\mathrm{supp}w$, so the assembled interpolant has compact support in $\mathrm{\Omega }$. Its zero extension $U$ is in $H(\mathrm{curl};{\mathbb{R}}^3)$: cellwise integration by parts cancels the interior face terms by [F3], and there is no boundary jump because $U$ vanishes near $\mathrm{\partial }\mathrm{\Omega }$. Choose $\rho$ as in [F5] with $r=1\mathrm{/}2$ and $R=1$, and put $c=({\int }_{{\mathbb{R}}^3}\rho {)}^{-1}$. Then $0<c<\mathrm{\infty }$ by [F6]. For $\eta >0$, set ${\kappa }_{\eta }(x)=c{\eta }^{-3}\rho (x\mathrm{/}\eta )$. This family is nonnegative, has unit integral and $L^1$ norm one, and is supported in $B_{\eta }(0)$; hence it is an $L^1$ approximate identity by An $L^1$ approximate identity on ${\mathbb{R}}^n$. Take ${\eta }_j=(d-h)\mathrm{/}(j+2)$, so ${\eta }_j<d-h$ and ${\eta }_j\to 0$. By [F8], ${\kappa }_{{\eta }_j}\ast U$ is smooth. The support of $U$ lies within distance $h$ of $\mathrm{supp}w$, so $\mathrm{dist}(\mathrm{supp}U,\mathrm{\partial }\mathrm{\Omega })\ge d-h$. Since $\mathrm{supp}{\kappa }_{{\eta }_j}\subset B_{{\eta }_j}(0)$, the convolution vanishes when $\mathrm{dist}(x,\mathrm{supp}U)>{\eta }_j$; its support is therefore in the closed ${\eta }_j$-neighborhood of $\mathrm{supp}U$, which is compactly contained in $\mathrm{\Omega }$ because ${\eta }_j<d-h$. Thus it is in $C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$. By [F7], its curl is ${\kappa }_{{\eta }_j}\ast (\mathrm{curl}U)$. The approximate-identity convergence [F4] applied to both $U$ and $\mathrm{curl}U$ gives convergence in the $H(\mathrm{curl})$ graph norm. Thus $I_{h}w\in H_0(\mathrm{curl};\mathrm{\Omega })$ by the closure definition.

By [F1], $\mathrm{curl}{I}_{K}w$ is a constant vector $c_K$ on $K$. For each oriented face $F\subset \mathrm{\partial }K$, [F3] and the defining edge moments give ${\int }_F{c}_K\cdot n_{F}dS={\int }_F\mathrm{curl}w\cdot n_{F}dS$. With $c_0=\mathrm{curl}w(x_K)$, smoothness bounds the difference between the right side and $\mathrm{\mid }F\mathrm{\mid }{c}_0\cdot n_F$ by $\mathrm{\mid }F\mathrm{\mid }{h}_K\mathrm{\parallel }D(\mathrm{curl}w){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$. Three face normals span ${\mathbb{R}}^3$. More precisely, choose three reference faces whose unit normals form a basis. Their physical normals are normalized cofactor images under $B_K^{-T}$; shape regularity bounds the condition number of $B_K\mathrm{/}{h}_K$, so the inverse of this three-normal matrix is uniformly bounded. Hence $\mathrm{\mid }{c}_K-c_0\mathrm{\mid }\le C{h}_K\mathrm{\parallel }D(\mathrm{curl}w){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$ and $\mathrm{\parallel }\mathrm{curl}{I}_{K}w-\mathrm{curl}w{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{5\mathrm{/}2}\mathrm{\parallel }D(\mathrm{curl}w){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}$.

Shape regularity gives $\mathrm{\mid }K\mathrm{\mid }\ge c{h}_K^3$, so ${\sum }_K{h}_K^5\le h^2{\sum }_K{h}_K^3\le C{h}^2\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$. Squaring the local bounds in steps 1.1 and 2.1, summing over the mesh, and taking square roots gives $\mathrm{\parallel }{I}_{h}w-w{\mathrm{\parallel }}_{H(\mathrm{curl};\mathrm{\Omega })}\le C_{w}h⟶0$. The constant depends on the fixed smooth field, the domain and the mesh shape-regularity bound, but not on $h$.