Direct numerical simulation of an L2 supercritical variant of the derivative nonlinear Schrödinger equation suggests that there is a finite time singularity. Subsequent exploration with the dynamic rescaling method provided more detail about the blowup and a recent refined asymptotic analysis of the blowup solution gives predictions of the blowup rates. However, due to the mixed hyperbolic-dispersive nature of the equation, these methods have limited the proximity to the blowup time. Using a locally adaptive meshing method, we are able to overcome these difficulties.