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 πh:Vh→Dh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _h: V_h \rightarrow D_h$$\end{document} of finite element approximation space Vh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_h$$\end{document} into a discontinuous space Dh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_h$$\end{document}. 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.