We present a method for simulation of viscoelastic flows using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation. In the second step the mesh is moved in ALE mode and the associated convection of the stress is performed using a Godunov type scheme. This ALE technique is easy to implement and can accurately simulate the complex viscoelastic behaviour of transient polymer flow through complex geometries. In the present study, steady flows through abrupt contractions of planar and axisymmetric geometries are studied by performing transient flow simulations until steady state is achieved. The proposed method is validated with previously published numerical and experimental studies for polymer solutions obeying the Oldroyd B and Phan Thien Tanner (PTT) models. The simulated corner and lip vortex enhancement mechanism and flow behavior are in good agreement with experimentally obtained flow visualization photographs. The strength of the proposed method lies in its ability to simulate free surface flows such as swell. (C) 2007 Elsevier B.V. All rights reserved.