numbers of vertices and elements. (Their generation will be discussed in the next subsection.) Denote the coordinates of the vertices of $\cTh^n$ by ${\boldsymbol{x}}_j^n$, $j = 1, ..., N_v$, where $N_v$ is the number of all vertices. The mesh $\cTh(t)$ between any two time instants $t_n$ and $t_{n+1}$ is defined through linear interpolation, i.e., $$\begin{aligned}
&    {\boldsymbol{x}}_j(t) =
    \frac{t-t_n}{t_{n+1}-t_n} {\boldsymbol{x}}_j^{n+1} +
    \frac{t_{n+1}-t}{t_{n+1}-t_n} {\boldsymbol{x}}_j^{n} , \quad \forall j = 1, ..., N_v 
\\
&\dot{{\boldsymbol{x}}}_j (t) = \frac{{\boldsymbol{x}}_j^{n+1}-{\boldsymbol{x}}_j^{n}}{t_{n+1}-t_n},\quad
j = 1, ..., N_v.\end{aligned}$$ Denote by $\phi_j({\boldsymbol{x}}, t)$ the linear basis function associated with vertex ${\boldsymbol{x}}_j(t)$. For convenience, we assume that the vertices are arranged such that the first $N_{vi}$ vertices are the interior vertices. Let $$V_h(t) = \text{span}\{\phi_1(\cdot, t), ..., \phi_{N_{vi}}(\cdot, t)\} .$$ Then, the linear finite element approximation to the solution of IBVP is defined as $u_h(\cdot, t) \in V_h(t)$, $ t \in (t_0, T]$ such that $$\begin{cases} 
    & \int_{\Omega} \frac{\p u_h}{\p t} v~d{\boldsymbol{x}} = - \int_{\Omega} |u_h|^m \grad u_h
    \cdot \grad v~d{\boldsymbol{x}} , 
        \quad \forall v \in V_h(t),\quad t_0 < t \leq T  \\
    & \int_{\Omega} (u_h({\boldsymbol{x}},0) - u^0({\boldsymbol{x}})) v~d{\boldsymbol{x}}  = 0, \quad \forall v \in V_h(t) .
    \end{cases}
    \label{fem0}$$ The above equation can be cast in matrix form. Indeed, 