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

The paper treats of the numerical approximation for the following boundary value problem: $$ left{ begin{gathered} u_t (x,t) - u_{xx} (x,t) = 0, 0 < x < 1, t in (0,T), hfill u(0,t) = 1, u_x (1,t) = - u^{ - p} (1,t), t in (0,T), hfill u(x,0) = u_0 (x) > 0, 0 leqslant x leqslant 1, hfill end{gathered} right. $$ where p > 0, u ₀ ∈ C ²([0, 1]), u ₀(0) = 1, and u′ ₀(1) = ?u ₀ ^?p (1). Conditions are specified under which the solution of a discrete form of the above problem quenches in a finite time, and we estimate its numerical quenching time. It is also proved that the numerical quenching time converges to real time as the mesh size goes to zero. Finally, numerical experiments are presented which illustrate our analysis.