\qquad\mbox{for any $\alpha\in(0,\delta_*]$} .\end{aligned}$$* ]{}

As a consequence of Lemma \[RegPoiss\], we have the following result on the regularity of ${\bf w}_j$, (which is also a consequence of Proposition 3.7 of [@ABDG], but for self-containedness we include a short proof here).

\[RegHarV\] [ *For any given curved polyhedron $\Omega$, there exists a positive constant $\delta_*>0$ such that the harmonic vector fields ${\bf w}_j$, $j=1,2,\cdots,\frak M$, are in ${\bf H}^{1/2+\delta_*}(\Omega)$.* ]{}

[*Proof of Lemma \[RegHarV\]*]{}.$\,\,\,$ Let $\Sigma_{j}'$ be a small perturbation of the surfaces $\Sigma_{j}$ for each $j=1,\cdots,\frak M$, such that $\Sigma_j'\cap\Sigma_k=\emptyset$ and $\Omega\backslash\Sigma'$ is simply connected (where $\Sigma'=\cup_{j=1}^{\frak M}\Sigma_j'$). Let $D_\Sigma$ and $D_\Sigma'$ be small neighborhoods of $\Sigma$ and $\Sigma'$, respectively, such that $\overline D_\Sigma\cap\overline D_\Sigma'=\emptyset$.

By using Lemma \[RegPoiss\] it is easy to show that the solution of satisfies $$\begin{aligned}
&\varphi_j\in H^{3/2+\delta_*}(\Omega\backslash\overline D_\Sigma) ,
\quad j=1,2,\cdots,\frak M ,\end{aligned}$$ which implies that ${\bf w}_j=\nabla\varphi_j$, $j=1,\cdots,\frak M$, are $H^{1/2+\delta_*}$ in the subdomain $\Omega\backslash\overline D_\Sigma$. Similarly, if we define $\varphi_j'$ as the solution of with $\Sigma_i$ replaced by $\Sigma_i'$, then ${\bf w}_j':=\nabla\varphi_j'$, $j=1,\cdots,\frak M$, also form a basis of ${\bf X}(\Omega)$, and they are $H^{1/2+\delta_*}$ in the subdomain $\Omega\backslash\overline D_\Sigma'$. Since ${\bf w}_j$ can be expressed as linear combinations of ${\bf w}_j'$, 