This paper concerns the study
of the
numerical approximation for the following boundary value problem
$$
\left\{
\begin{array}{ll}
\hbox{$(u^{m})_{t}=u_{xx}$, $0$,} \\
\hbox{$u_{x}(0,t)=0$,\quad $u_{x}(1,t)=-u^{-\beta}(1,t)$,\quad $t>0$,} \\
\hbox{$u(x,0)=u_{0}(x)>0$,\quad $0\leq x\leq 1$,} \\
\end{array}
\right.$$
where $m\geq1$, $\beta>0$. We obtain some conditions under
which the solution of a semidiscrete form of the above problem
quenches in a finite time and estimate its semidiscrete quenching
time. We also establish the convergence of the semidiscrete
quenching time. Finally, we give some numerical experiments to
illustrate our analysis.