An Upwind Weak Galerkin Scheme for Convection-Dominated Oseen Equations

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 \(k\geqslant 1\) $k⩾1$ , the numerical solutions are proved to converge at the rate of \(O(h^{k+\frac{1}{2}})\) $O(h^{k+\frac{1}{2}})$ 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.