Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions (Q1599872)

The authors considered the asymptotic behavior of a semidiscrete numerical approximation for the heat equation in a bounded smooth domain with nonlinear flux condition \(\frac{\partial u}{\partial \eta}=u^p\). They proved that every numerical solution blows up in finite time iff \(p > 1\) and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero. The blow-up rate for the numerical is different from the continuous one. Moreover they find that the blow-up set for the numerical solution is contained in a small neighborhood of the blow-up set of the continuous problem when the mesh parameter is small enough.