(see Figure \[FigD\]) .

The integer $\frak M$ is often referred to as the first Betti number of the domain. The existence of the surfaces $\Sigma_j$, $j=1,\cdots,\frak M$, is only needed in the analysis of the finite element solutions by using the Hodge decomposition [@KY09]. One does not need to know these surfaces in practical computation.

[![Illustration of the domain [(]{}$\Omega$ is the shadow region[)]{}.[]{data-label="FigD"}](Fig1.eps "fig:"){height="1.3in" width="1.5in"}]{} [![Illustration of the domain [(]{}$\Omega$ is the shadow region[)]{}.[]{data-label="FigD"}](Fig2.eps "fig:"){height="1.3in" width="1.5in"}]{}

[**Assumptions 2.1.**]{}$\,\,\,$ We assume that $\Omega\subset\R^3$ is a curved polyhedron which is partitioned into quasi-uniform tetrahedra. For any given integers $$\begin{aligned}
\label{Condrk}
r\geq 1 
\quad\mbox{and}\quad
k\geq 2r-1 ,\end{aligned}$$ we denote by ${\mathbb S}_h^{r}$ the complex-valued Lagrange finite element space of degree $\leq r$, denote by ${\mathbb V}_h^{k+1}$ the real-valued Lagrange finite element space of degree $\leq k+1$, and let ${\mathbb N}_h^k$ be either the Nédélec 1st-kind H(curl) element space of order $k$ [@Ned80] or the Nédélec 2nd-kind H(curl) element space of degree $\leq k$ [@Ned86] (also see page 60 of [@AFW]).

Let the time interval $[0,T]$ be partitioned into $0=t_0<t_1<\cdots<t_N$ uniformly, with $\tau=t_{n+1}-t_n$. For any given functions $f_n$, $n=0,1,\cdots,N$, we define its discrete time derivative as $$\begin{aligned}
D_\tau f^{n+1}:=(f^{n+1}-f^n)/\tau ,\quad
n=0,1,\cdots,N-1 .\end{aligned}$$ 