,  \label{PDE_hodge_u3}\end{aligned}$$ $\omega$ is the solution of $$\begin{aligned}
\label{Eqomega}
&\Delta \omega=\nabla\cdot {\bf v}
&&\mbox{in}\,\,\,\Omega,  {\nonumber}\\
&\nabla \omega\cdot{\bf n}={\bf v}\cdot{\bf n}
&&\mbox{on}\,\,\, \partial\Omega,\end{aligned}$$ and ${\bf w}_j=\nabla\varphi_j$, $j=1,2,\cdots,\frak M$, form a basis for ${\bf X}(\Omega)$ with $\varphi_j$ being the solution of $$\begin{aligned}
\label{DHarmF}
&\Delta\varphi_j=0 &&\mbox{in}\,\,\,\Omega\backslash\Sigma, {\nonumber}\\
&\nabla \varphi_j\cdot{\bf n}=0
&&\mbox{on}\,\,\, \partial\Omega,  \\
&[\nabla \varphi_j\cdot{\bf n}]=0
\quad\mbox{and}\quad [\varphi_j]=\delta_{ij}
&&\mbox{on}\,\,\, \Sigma_i ,\quad i=1,2,\cdots,\frak M,  {\nonumber}\end{aligned}$$ ($\delta_{ij}$ denotes the Kronecker symbol). The coefficients $\alpha_j$, $j=1,\cdots,\frak M$, are given by $$\begin{aligned}
\label{DefAlphaj}
\alpha_j=({\bf v},{\bf w}_j)/\|{\bf w}_j\|_{L^2}^2 . \end{aligned}$$

Although $\varphi_j$ is only defined on $\Omega\backslash\Sigma$, the gradient $\nabla\varphi_j$ has a natural extension to be a vector field in ${\bf H}({\rm curl,div})$ due to the interface conditions.

To study the regularity of ${\bf w}_j$, we cite the following lemma on the regularity of the Poisson equation in a polyhedral domain. This result can be obtained by substituting fractional $k$ in Corollary 3.9 of [@Dauge] (also see page 30 of [@Dauge08] and (23.3) of [@Dauge88]).

\[RegPoiss\] [*For any given curved polyhedron $\Omega$, there exists a positive constant $\delta_*>0$ such that the solution of the Poisson equation $$\begin{aligned}
\left\{\begin{array}{ll}
-\Delta \varphi=f
&\mbox{in}\,\,\,\Omega, \\
\nabla\varphi\cdot{\bf n}=0
&\mbox{on}\,\,\,\partial\Omega,
\end{array}\right.  \end{aligned}$$ with the normalization condition $\int_\Omega\varphi\d x=0$, satisfies $$\begin{aligned}
\|\varphi\|_{H^{3/2+\alpha}(\Omega)} 
\leq C\|f\|_{H^{-1/2+\alpha}(\Omega)}  
