it follows that ${\bf w}_j$ is $H^{1/2+\delta_*}$ in the subdomain $\Omega\backslash \overline{D_\Sigma'}\supset\overline D_\Sigma$. Therefore, ${\bf w}_j$ is $H^{1/2+\delta_*}$ in the whole domain $\Omega$.

[ *We define the following finite element subspaces of ${\mathbb N}_h^k\subset {\bf H}({\rm curl})$: $$\begin{aligned}
&{\bf C}_h(\Omega):=\{
{\bf v}_h\in {\mathbb N}_h^k:\,
\nabla\times{\bf v}_h=0 \} , \\
&
{\bf G}_h(\Omega):=\{
\nabla\chi_h:\,
\chi_h\in {\mathbb V}_h^{k+1}\} , \\
&{\bf X}_h(\Omega):=\{
{\bf v}_h\in {\mathbb N}_h^k:\,
\nabla\times {\bf v}_h=0,\,\,
({\bf v}_h,\nabla\chi_h)=0,\,\,
\forall\chi_h\in {\mathbb V}_h^{k+1}\}\end{aligned}$$ where ${\bf X}_h(\Omega)$ is often referred to as the space of discrete harmonic vector fields.* ]{}

With the notations above, we have the discrete Hodge decomposition (page 72 of [@AFW]): $$\begin{aligned}
\label{DHodgeD}
{\mathbb N}_h^k={\bf C}_h(\Omega)^\perp
\oplus{\bf G}_h(\Omega)\oplus {\bf X}_h(\Omega)  .\end{aligned}$$ The following lemma is concerned with the regularity of the discrete harmonic vector fields.

\[RegDHarV\] [*For any given curved polyhedron $\Omega$, there exists a positive constant $h_0$ such that when $h<h_0$ the space ${\bf X}_h(\Omega)$ has an orthogonal basis $\{{\bf w}_{j,h}$: $j=1,\cdots,\frak M\}$ which satisfies $$\begin{aligned}
&\sum_{j=1}^{\frak M}\|{\bf w}_{j,h}\|_{L^{3+\delta}}\leq C 
\qquad\mbox{and}\qquad
\sum_{j=1}^{\frak M}\|{\bf w}_{j,h}-{\bf w}_j\|_{L^{3+\delta}}
\rightarrow 0 \quad\mbox{as}\,\,\, h\rightarrow 0  , \end{aligned}$$ for any $0<\delta<3\delta_*/(1-\delta_*)$, where $\delta_*$ is given by Lemma \[RegHarV\].* ]{}

[*Proof of Lemma \[RegDHarV\]*]{}.$\,\,\,$ If ${\bf v}_h\in {\bf X}_h(\Omega)$, then $\nabla\times {\bf v}_h=0$ and so 