Numerical approximation of a parabolic problem with a nonlinear boundary condition

We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an approximation of the original problem. Our objective is to analyze whether this system has a similar behaviour than the original problem. We find a necessary and sufficient condition for blow up of this system. However, this condition is slightly different than the one known for the original problem, in particular, there are cases in which the continuous problem has blow up while its semidiscrete approximation does not.
Under certain assumptions we also prove that the numerical blow up time converges to the real blow-up time when the meshsize goes to zero. Our proofs are given in one space dimension. Similar arguments could be applied for higher dimensions but a further analysis is required.