Numerical Approximation of a Parabolic Problem with a Nonlinear Boundary Condition in Several Space Dimensions

In this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for the heat equation, ut = ∆u, in a bounded smooth domain, with a nonlinear flux boundary condition at the boundary, ∂u ∂η = up. We focus in the behaviour of blowing up solutions. First we prove that every numerical solution blows up in finite time if and only if p > 1 and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero. Next, we show that the blow-up rate for the numerical scheme is different from the continuous one. Nevertheless we find that the blow-up set for the numerical approximations it is contained in a neighborhood of the blow-up set of the continuous problem, when the mesh parameter is small enough.