A pressure-based finite volume method on unstructured grids is developed for incompressible viscoelastic fluid flow simulations. To avoid the need for information about the stresses at any singular point within the computational domain, an edge (or face for 3D cases) based formulation with all variables located at the cell centroid is used. To avoid possible decoupling between the stress and velocity fields, and ensure consistencies in discretization under the collocated variable arrangement, the deformation rates and stress tensors at the midpoint of an interface straddled by two adjacent cells are linearly interpolated with the variable gradients expressed in terms of the values at all neighboring cell centroids using a least-squares method. An innovative solution strategy that removes the need for explicitly specifying the flow boundary conditions at an open boundary, originally developed for incompressible Newtonian flow simulations, has been extended for viscoelastic cases. With this method, the traditional requirement for flow conditions at an outflow boundary, i.e. fully developed flow, is removed. The outflow channel can thus be truncated so that the flow outlet can be located arbitrarily as long as there is no inflow (e.g. due to recirculation) across this boundary. This not only greatly reduces computational costs, but avoids numerical instabilities due to inconsistencies between the imposed flow behavior and that calculated at the outflow boundary for cases where the outflow channel is not long enough to realize the flow condition imposed. The proposed methodology is demonstrated/validated by simulating, developing flow of an Oldroyd-B fluid between two parallel plates, and slip-stick flow of an upper convected Maxwell fluid; with successively truncated flow domains. These results clearly demonstrate that the predicted flow behavior is unaffected by using even a highly truncated domain, whilst this methodology allows a higher critical Weissenberg number to be reached than previously reported in the literature. (C) 2016 Elsevier B.V. All rights reserved.