Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions
Gabriel Acosta, Julián Fernández Bonder, Pablo Groisman, Julio Daniel Rossi · Discrete and Continuous Dynamical Systems - B · 2002
In this paper we study the asymptotic behavior of a semidiscretenumerical approximation for the heat equation, $u_t = \Delta u$, in a bounded smoothdomain with a nonlinear flux boundary condition, $(\partial u)/(\partial\eta)= u^p$. We focus in thebehavior of blowing up solutions. We prove that every numerical solutionblows up in finite time if and only if $p > 1$ and that the numerical blow-uptime converges to the continuous one as the mesh parameter goes to zero. Alsowe show that the blow-up rate for the numerical scheme is different from thecontinuous one. Nevertheless we find that the blow-up set for the numericalapproximations is contained in a small neighborhood of the blow-up set of thecontinuous problem when the mesh parameter is small enough.