Adaptive numerical schemes for a parabolic problem with blow-up

In this paper we present adaptive procedures for the numerical study of positive solutions of the following problem: ut = uxx (x,t) ∈ (0,1) x [0,T), ux(0,t) = 0 t ∈ [0,T], ux(1,t) = up(1,t) t ∈ [0, T), u(x,0) = u0(x) x ∈ (0, 1), with p > 1. We describe two methods. The first one refines the mesh in the region where the solution becomes bigger in a precise way that allows us to recover the blow-up rate and the blow-up set of the continuous problem. The second one combines the ideas used in the first one with moving mesh methods and moves the last points when necessary. This scheme also recovers the blow-up rate and set. Finally, we present numerical experiments to illustrate the behaviour of both methods.