A new numerical method specially adapted to the free-surface flow of viscoelastic material is proposed. It is based on a Lagrangian discretisation of the material and objective derivatives, which accounts well for the hyperbolic nature of these terms and goes well with a Lagrangian tracking of a time-evolving domain. Through the Arbitrary Lagrangian-Eulerian (ALE) formulation, the method can also be applied efficiently to solid-boundary problems, and is tested on the benchmark problem of the drag on a cylinder in a channel. The collapse of a column of Oldroyd-B fluid is then considered: under the action of surface tension, the column undergoes large deformation leading to the "beads-on-string" structure. Asymptotic results on the evolution of this structure are recovered in numerical simulations and further features of this flow are exhibited. (c) 2006 Elsevier B.V. All rights reserved.