{\bf G}(\Omega)\oplus 
{\bf X}(\Omega) , \end{aligned}$$ where $$\begin{aligned}
&{\bf C}(\Omega):=\{{\bf u}\in {\bf H}({\rm curl}): \nabla\times{\bf u} =0\}, \\
&{\bf C}(\Omega)^\perp=\{\nabla\times{\bf u}: {\bf u}\in {\bf H}({\rm curl}),\,\,{\bf u}\times{\bf n}=0\} ,\\
&{\bf G}(\Omega):=\{\nabla\omega: \omega\in H^1\} , \\
&{\bf X}(\Omega):=\{{\bf w}\in {\bf H}({\rm curl})\cap{\bf H}({\rm div}):\,
\nabla\times{\bf w}=0, \,\,
\nabla\cdot{\bf w}=0 ,\,\, {\bf w}\cdot{\bf n}=0\,\,\,\mbox{on}
\,\,\,\partial\Omega\}  ,\end{aligned}$$ ${\bf C}(\Omega)^\perp$ denotes the orthogonal complement of ${\bf C}(\Omega)$ in ${\bf L}^2$, and ${\bf X}(\Omega)$ is the space of harmonic vector fields.

The second type of space of harmonic vector fields is defined by [^6] $$\begin{aligned}
\widetilde {\bf X}(\Omega):=\{{\bf w}\in {\bf H}({\rm curl})\cap{\bf H}({\rm div}): \nabla\times{\bf w}=0,\,\,
\nabla\cdot{\bf w}=0
\,\,\mbox{and}\,\, {\bf w}\times {\bf n}=0
\,\,
\mbox{on}\,\,\partial\Omega\} ,\end{aligned}$$ and we denote $$\begin{aligned}
\label{Def_Space_Y}
\widetilde {\bf Y}(\Omega):=\{{\bf w}\in {\bf H}({\rm curl})\cap{\bf H}({\rm div})\cap 
\widetilde {\bf X}(\Omega)^\perp: {\bf w}\times {\bf n}=0\,\,
\mbox{on}\,\,\partial\Omega \} .\end{aligned}$$ As a result of , any vector field ${\bf v}\in {\bf L}^2$ has the Hodge decomposition (also see [@KY09 Appendix]) $$\begin{aligned}
\label{HodgeDM}
{\bf v} =
\nabla\times {\bf u} +\nabla \omega 
+\sum_{j=1}^{\frak M}\alpha_j{\bf w}_j ,\end{aligned}$$ where ${\bf u}\in \widetilde {\bf Y}(\Omega)$ is the solution of the problem [^7] [^8] $$\begin{aligned}
&\nabla\times(\nabla\times {\bf u})=
\nabla\times {\bf v} &&\mbox{in}\,\,\,\Omega,  \label{PDE_hodge_u1}\\
&\nabla\cdot{\bf u}=0 &&\mbox{in}\,\,\,\Omega,\\
&{\bf u}\times {\bf n}=0
&&\mbox{on}\,\,\,\partial\Omega 