Stabilized lowest equal-order mixed finite element method for the Oseen viscoelastic fluid flow

In this paper, we present a stabilized lowest equal-order mixed finite element (FE) method for the Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law. To approximate the velocity, pressure, and stress tensor, we choose lowest equal-order FE triples p1−p1−p1dg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p1-p1-p1_{\mathrm{dg}}$\end{document} respectively. It is well known that these elements don’t satisfy the inf–sup (or LBB) condition. Owing to the violation of the essential stability condition, the system became unstable. To overcome this difficulty, a standard pressure stabilization term is added to the discrete variational formulation, which ensures the well-posedness of the FE scheme. The existences and uniqueness of the FE scheme are derived. The desired optimal error bound is obtained. Three numerical experiments are executed to illustrate the validity and efficiency of the numerical method. The stabilized method provides attractive computational advantages, such as simpler data structures, parameter-free, no calculations of higher-order derivatives, and fast solver in simulations.