Robust and Globally Divergence-Free Weak Galerkin Methods for Oseen Equations

Abstract

In this paper, a robust and globally divergence-free weak Galerkin finite element method of Oseen equations is proposed and analyzed. We use the $\mathbf{P}_{k}/P_{k-1}$ discontinuous finite element combination for the approximation of velocity and pressure, and piecewise $\mathbf{P}_{k}/P_{k}$ for the numerical traces of velocity and pressure. This method not only yields globally divergence-free velocity approximations, but is also robust in the sense that a priori error estimates are uniform with respect to the coefficients of Oseen equations, providing the exact solutions are sufficiently smooth. Finally, numerical examples are given to confirm our theoretical results.