Published |
ABSTRACT:
In this paper, we consider the use of the $H(\div)$ element in the velocity-pressure formulation to discretize Stokes equations. We address the error estimate of the element pair $\rm{RT}_0$-$\rm{P}_0$, which is known to be suboptimal, and render the error estimate optimal by the symmetry of the grids and by the superconvergence result of Lagrange interpolant. By enlarging $\rm{RT}_0$ such that it becomes a modified $ \rm{BDM}$-type element, we develop a new discretization ${\rm BDM}_1^{\rm b}$-${\rm P}_0$. We, therefore, generalize the classical MAC scheme on rectangular grids to triangular grids and retains all the desirable properties of the MAC scheme: exact divergence-free, solver-friendly, and local conservation of physical quantities. Further, we prove that the proposed discretization achieves the optimal convergence rate for both velocity and pressure on general quasi-uniform unstructured grids, and one and half order convergence rate for the vorticity and a recovered pressure. We demonstrate the validity of theories developed here by numerical experiments.