The fast stretching of fluid filaments to form either drops or broken threads is sensitive to both fluid rheology and surface tension and in this paper numerical tools are used to simulate the behaviour and compare simulations with experimental results. The simulation uses finite element techniques and the free surface is tracked using a level set method. By using mesh adaptation it was possible to simulate both Newtonian and yield stress fluids during rapid stretching between moving pistons and capture the deformation, breakup and final form behaviour in the ms time scales. The simulations are able to show clearly how Newtonian viscosity, surface tension and Bingham yield stress all influence the deformation in different ways. Initially, the simulations were tested against matching experimental observations for low and high viscosity Newtonian fluids and both the experimentally observed end pinching for low viscosity fluids and the longer time scale linear filament thinning for the higher viscosity fluid were successfully matched with the experimental results. A further selection of complex fluids with yield stress characteristics was also experimentally tested and the simulation matched using a Bingham and or Carreau-Yasuda constitutive equation. By suitable adjustment of the constitutive parameters to match experimental stress sweep rheological data, the filament deformation simulations were in general able to give realistic agreement with the experimental observations.