A finite element model is presented to describe the flow, resulting stresses and crystallization in a filament stretching extensional rheometer (FiSER). This model incorporates nonlinear viscoelasticity, nonisothermal processes due to heat release originating from crystallization and viscous dissipation as well as the effect of crystallization on the rheological behavior. To apply a uniaxial extension with constant extension rate, the FiSER plate speed is continuously adjusted via a radius-based controller. The onset of crystallization during filament stretching is investigated in detail. Even before crystallization starts, the rheology of the material can change due to the effects of flow-induced nucleation on the relaxation times. Both nucleation and structure formation are found to be strongly dependent on temperature, strain rate and sample aspect ratio. The latter dependence is caused by a clear distribution of crystallinity over the radius of the filament, which is a result of the nonhomogeneous flow history in the FiSER. Therefore, this numerical model opens the possibility to a priori determine sample geometries resulting in a homogeneous crystallinity or to account for the nonhomogeneity.