A new three-field formulation based on the Local Projection Stabilization (LPS) is developed for computations of the coupled Navier-Stokes and Oldroyd-B viscoelastic constitutive equations at high Weissenberg numbers. One-level LPS is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh. It allows us to use equal order interpolation spaces for the velocity and the viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and the pressure. Further, the coupled system of equations are solved in a monolithic approach. Since the stabilization terms in LPS are assembled only once, the proposed scheme is computationally efficient in comparison with residual based stabilized numerical schemes. Numerical studies using method of manufactured solutions show an optimal order of convergence in the respective norms. Further, the proposed scheme is validated using two benchmark problems: flow past a cylinder in a rectangular channel and lid-driven cavity flow. Moreover, the numerical results are compared with the results in the literature and the effects of elasticity and inertia are studied. (C) 2017 Elsevier B.V. All rights reserved.