the Hodge decomposition implies $$\begin{aligned}
 {\bf v}_h =
\nabla \omega 
+\sum_{j=1}^{\frak M}\alpha_j{\bf w}_j .\end{aligned}$$ Using the commuting property of the smoothed projection operator (Lemma \[SmoothPr\]) we derive $$\begin{aligned}
 \label{DPvh}
{\bf v}_h 
&= \widetilde\Pi_h^{\mathbb N}\nabla \omega 
+\sum_{j=1}^{\frak M}\alpha_j\widetilde\Pi_h^{\mathbb N} {\bf w}_j 
= \nabla \widetilde\Pi_h^{\mathbb V}\omega 
+\sum_{j=1}^{\frak M}\alpha_j\widetilde\Pi_h^{\mathbb N} {\bf w}_j 
=: \nabla \omega_h 
+\sum_{j=1}^{\frak M}\alpha_j\widetilde\Pi_h^{\mathbb N} {\bf w}_j ,\end{aligned}$$ where we have defined $\omega_h:=\widetilde\Pi_h^{\mathbb V}\omega$ to simplify the notation. Since any ${\bf v}_h\in {\bf X}_h(\Omega)$ satisfies $({\bf v}_h,\nabla\chi_h)=0$ for all $\chi_h\in {\mathbb V}_h^{k+1}$, it follows that $$\begin{aligned}
(\nabla \omega_h,\nabla\chi_h)
=-\sum_{j=1}^{\frak M}\alpha_j
(\widetilde\Pi_h^{\mathbb N} {\bf w}_j,\nabla\chi_h)
=\sum_{j=1}^{\frak M}\alpha_j
({\bf w}_j-\widetilde\Pi_h^{\mathbb N} {\bf w}_j,\nabla\chi_h),
\quad\forall\, \chi_h\in {\mathbb V}_h^{k+1} .\end{aligned}$$

If we define $\omega_{j,h}\in{\mathbb V}_h^{k+1}$ (with the normalization $\int_\Omega\omega_{j,h}\d x=0$) as the finite element solution of $$\begin{aligned}
\label{omegajh}
(\nabla \omega_{j,h},\nabla\chi_h)
=({\bf w}_j-\widetilde\Pi_h^{\mathbb N} {\bf w}_j,\nabla\chi_h),
\quad\forall\, \chi_h\in {\mathbb V}_h^{k+1} ,\end{aligned}$$ then we have $
\omega_{h}=\sum_{j=1}^{\frak M}
\alpha_j \omega_{j,h}  
+{\rm const} .
$ Substituting this into , we obtain $$\begin{aligned}
{\bf v}_h
=\sum_{j=1}^{\frak M}\alpha_j(\nabla \omega_{j,h} 
+ \widetilde\Pi_h^{\mathbb N} {\bf w}_j) .\end{aligned}$$ We see that any vector field in ${\bf X}_h(\Omega)$ can be expressed as a linear combination of $$\begin{aligned}
{\bf w}_{j,h}:=\nabla\omega_{j,h} 
+\widetilde\Pi_h^{\mathbb N} {\bf w}_j ,\quad
j=1,\cdots,\frak M .\end{aligned}$$ The vector fields ${\bf w}_{j,h}$, $j=1,\cdots,\frak 