\left\{\begin{array}{ll}
\partial_tu-\Delta u =f  & \mbox{in}\,\,\,\Omega ,\\
\nabla u\cdot{\bf n}=0 & \mbox{on}\,\,\,\partial\Omega ,\\
u(x,0)=0, & \mbox{for}\,\,x\in\Omega ,
\end{array}\right. \end{aligned}$$ possesses the maximal $L^p$-regularity (see Corollary 4.d of [@Weis1]): $$\begin{aligned}
&\|\partial_tu\|_{L^p(0,T;L^q)}
+\|\Delta u\|_{L^p(0,T;L^q)}
\leq C_{p,q}\|f\|_{L^p(0,T;L^q)} ,\qquad 
1<p,q<\infty .\end{aligned}$$ In this paper, we need to use the maximal $\ell^p$-regularity for time-discrete parabolic PDEs, which was proved in [@KLL Theorem 3.1].

\[DMPR\][ *The solution of the time-discrete PDEs $$\begin{aligned}
\left\{\begin{array}{ll}
D_\tau u^{n+1}-\Delta u^{n+1}=f^{n+1}  & \mbox{in}\,\,\,\Omega ,\\
\nabla u^{n+1}\cdot{\bf n}=0 & \mbox{on}\,\,\,\partial\Omega ,\\
u^0=0,
\end{array}\right. \end{aligned}$$ $n=0,1,\cdots$, satisfies $$\begin{aligned}
&\bigg(\sum_{n=0}^m\tau\|D_\tau u^{n+1}\|_{L^q}^p\bigg)^{\frac{1}{p}}
+\bigg(\sum_{n=0}^m\tau\|\Delta u^{n+1}\|_{L^q}^p\bigg)^{\frac{1}{p}} 
\leq C_{p,q}\bigg(\sum_{n=0}^m\tau\|f^{n+1}\|_{L^q}^p\bigg)^{\frac{1}{p}}\end{aligned}$$ for any $1<p,q<\infty$ and $m\geq 0$, where the constant $C_{p,q}$ is independent of $\tau$ and $m$.* ]{}

We introduce some lemmas in Section \[DisHodg\] on the discrete Hodge decomposition, with emphasis on the uniform regularity of the discrete harmonic functions in curved polyhedra. A discrete Sobolev embedding inequality for functions in the Nédélec element space is proved in Section \[SobNed\]. With these mathematical tools, we present estimates and prove compactness/convergence of the finite element solution in Section \[CompFES\].

Discrete Hodge decomposition and harmonic vector fields {#DisHodg}
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It is well known that the following Hodge decompositions holds (for example, see [@AFW decomposition (2.18)]) [^5] $$\begin{aligned}
\label{Hodge_D_L2}
&{\bf L}^2={\bf C}(\Omega)^\perp\oplus 
