A stabilized local projection finite element scheme for computations of oldroyd-B viscoelastic fluid flows

This paper presents the numerical analysis of the three-field stabilized formulation based on the one-level local projection stabilization (LPS) for computations of the coupled Navier-Stokes and Oldroyd-B viscoelastic constitutive equations. Due to dominating convective terms, the velocity-pressure-stress formulation suffers from numerical instability in viscoelastic flows. The other challenges are the necessity of the inf-sup conditions for the velocity-pressure and stress-velocity couplings in equal-order interpolations. One-level local projection stabilization scheme 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 approximations. The local projection method is based on a projection pi(h) : V-h -> D-h of finite element approximation space V-h into a discontinuous space D-h. In one-level LPS, the approximation and projection spaces are defined on the same mesh, with an enriched approximation space. We prove the stability and a priori error analysis, ensuring the optimal order of convergence of the proposed numerical scheme. The numerical result validates the theoretical estimates.