This study constitutes a computational investigation of filament stretching at high Hencky-strains. It extends previous predictions on the multi-mode side through advance in Hencky-strain and single-mode predictions via theological variation, with the additional consideration of linear versus exponential stretching configurations using the Oldroyd-B model. The study focuses on strain-hardening polymeric solutions and widening the application range of an Arbitrary Lagrangian-Eulerian (ALE)-formulation hybrid finite element/volume scheme. Computational predictions are validated for consistency against theoretical solutions and through discrete refinement. Dominance of tension-thickening over shear-thinning properties is established for such strain-hardening fluids. For a Giesekus model and when shear stress intensifies, greater foot pinching results with filament mid-plane thickening; when hardening is reduced less extension is generated at the filament centre, which leads to more exaggerated thinning. We have been able to compare and contrast single versus multi-mode modelling (multi-timescale), pointing to the importance of each mode. The influence of multi-mode representation is made apparent through the developing components of stress generated and their impact upon filament shape. In generating solutions at large levels of Hencky-strain, we have been able to explore the occurrence of bead-like structures. We are able to identify that these are absent in either Giesekus or linear Phan-Thien/Tanner (xi = 0) solutions to Hencky-strains of the order of 5 units. In contrast to the linear Phan-Thien/Tanner (xi = 0.13) option, the presence of such structures is detected, at sufficiently high Hencky-strains above 3 units. For example, this occurs when body forces are included at 3.2 Hencky-strain units. (C) 2007 Elsevier B.V. All rights reserved.