A calculation method for two-dimensional time-dependent free surface flows is presented. The method extends the capabilities of CONNFFESSIT (calculation of non-Newtonian flows: finite elements and stochastic simulation technique) to transient free surface flows. The macroscopic unknowns are the velocity and the pressure fields together with the shape of the free surface. Particle tracking techniques, one of the main components of CONNFFESSIT, are capitalized upon in order to efficiently track the time evolution of the free surface. The ability of CONNFFESSIT to treat models for which no closed-form constitutive equation can be derived and to yield full molecular information is now enhanced with a technique for the calculation of free surface flows. In order to validate the method, transient, free jet swell calculations for the Oldroyd-B fluid using CONNFFESSIT are compared with continuum-mechanics POLYFLOW calculations. (C) 2001 The Society of Rheology.