parameter $\epsilon > 0$) and taking $\epsilon = \mathcal{O}(h^{\frac{2m+4}{m^2+4m+2}})$, Rose [@Rose-1983] shows that the error for a P1 finite element (for space) – backward Euler (for time) approximation of PME is bounded by $$\left (\sum_{n} \Delta t \| u_h^n - u \|_{L^{m+2}(\Omega)}^{m+2} \right )^{\frac{1}{m+2}}
\le C \left (\Delta t^{\frac{1}{m+1}} + \left (\ln \left (\frac{1}{h}\right )\right )^{\frac{1}{(m+1)(m+2)}} h^{\frac{2}{m+1}}\right ),
\label{rose-1}$$ where $h$ is the maximum element diameter and $u_h^n$ is the numerical approximation of $u$ at $t= t_n$. Nochetto and Verdi [@Nochetto-1988] consider a class of degenerate PDEs which includes PME as a special example and improve the result of [@Rose-1983]. They show that the error for a P1 finite element – 1st-order semi-implicit approximation of PME is bounded by $$\begin{aligned}
&\| u_h - u \|_{L^{\infty}(0,T; H^{-1}(\Omega))}
+ \| (u_{h})^{m+1} - u^{m+1} \|_{L^2(0,T; L^2(\Omega))}
+ \left \| \int_0^t ((u_{h})^{m+1} - u^{m+1})\right \|_{L^{\infty}(0,T; H^{1}(\Omega))}
\notag
\\
& \qquad \qquad 
\le C \left (\frac{h^2}{\epsilon} + \frac{h^4}{\epsilon^2 \Delta t} + \Delta t\right )^{\frac 1 2}
\label{nochetto-1}
\\
& \qquad \qquad = \mathcal{O} (h^{\frac{m+2}{2(m+1)}}),\quad \text{if}\quad 
\Delta t = \mathcal{O}(h^{\frac{m+2}{m+1}}),\quad \epsilon = \mathcal{O}(h^{\frac{m}{m+1}}).
\notag\end{aligned}$$ For the P1 finite element – backward Euler approximation, error bounds in various norm are obtained, for instance, $$\| u_h - u 