An adaptive scheme to treat the phenomenon of quenching for a heat equation with nonlinear boundary conditions

This paper concerns the study of numerical approximation for the following boundary value problem
$$ \begin{cases} u_t(x,t)-u_{xx}(x,t)=0,\quad 0<x<1,\ t\in(0,T),\\ u(0,t)=1,\ u_x(1,t)=-u^{-p}(1,t),\quad t\in(0,T),\\ u(x,0)=u_0(x)>0,\quad 0\le x\le 1, \end{cases} $$
where $p>0$, $u_0\in C^2([0,1])$, $u_0(0)=1$ and $u_0'(1)=-u_0^{-p}(1)$. We find some conditions under which the solution of a discrete form of the above problem quenches in a finite time and estimate its numerical quenching time. We also prove that the numerical quenching time converges to the real one when the mesh size goes to zero. Finally, we give some numerical experiments to illustrate our analysis.