An Upwind Weak Galerkin Scheme for Convection-Dominated Oseen Equations

Abstract

An upwind weak Galerkin finite element scheme was devised and analyzed in this article for convection-dominated Oseen equations. The numerical algorithm was based on the weak Galerkin method enhanced by upwind stabilization. The resulting finite element scheme uses equal-order, say k, polynomial spaces on each element for the velocity and the pressure unknowns. With finite elements of order

k⩾1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\geqslant 1$$\end{document}

, the numerical solutions are proved to converge at the rate of

O(h^{k+\frac{1}{2}})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(h^{k+\frac{1}{2}})$$\end{document}

in an energy-like norm for convection-dominated Oseen equations. Numerical results are presented to demonstrate the accuracy and effectiveness of the upwind weak Galerkin scheme.

Keywords

Convection-dominated / Weak Galerkin / Oseen equations / Upwind schemes

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