document: An adaptive time step procedure for a parabolic problem with blow-up In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time . We use standard linear elements with mass lumping for the space variable . For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then , we use explicit Runge-Kutta methods for this equivalent problem . In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case . We obtain necessary and sufficient conditions for the blowup of the numerical solution and prove that the numerical blow-up time converges to the continuous one . We also study , for the explicit Euler approximation , the localization of blow-up points for the numerical scheme
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