A blow up problem of the form \[ u_t(x,t)=(u^m)_{xx}(x,t),\;\;(x,t)\in (0,1)\times[0,T) \]
\[ (u^m)_x(0,t)=0, \;\;t \in [0,T), \]
\[ (u^m)_x(1,t)=u^p(1,t), \;\;t \in [0,T), \]
\[ u(x,0)=u_0(x), \;\;x \in (0,1), \] with \(p,m >0\) is investigated from the numerical point of view. Theoretical results concerning continuous problems for the porous medium equation (\( m>0\)) or the fast diffusion equation (\( 0 <m\leq 1\)) are well known, that means blow up time, blow up rate and blow up sets are derived for appropriate values of \(m\) and \(p\). Numerical approximations for these problems have been studied in the past using semidiscrete numerical approximation in a fixed mesh of size \(h\). But in comparing the behavior of continuous and numerical solutions significant differences have appeared. So the use of a fixed grid is not well suited for this problem.
In this paper the fully discrete numerical scheme is presented and its numerical solution obtains similar behavior as the continuous problem. The authors use an explicit Euler method with adaptive time step, but also the implicit Euler scheme and second order Runge-Kutta methods are briefly mentioned. For space discretization linear finite elements with mass lumping are used, and two methods are derived: the first method is a method of adding points between the last two mesh points near the blow up boundary; the second one is based on the idea of moving points near the boundary \(x=1\). For these two methods a uniform convergence result is obtained. Moreover it is proved that numerical blow up has the same range of exponents as the solution of the continuous problem, and also numerical blow up time converges to the continuous one. Finally the authors show that the blow up set of the numerical solution converges to the continuous blow up set.