Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

In this paper, we consider the use of $$H(\mathrm{div })$$H(div) elements in the velocity–pressure formulation to discretize Stokes equations in two dimensions. We address the error estimate of the element pair $$\mathrm{RT}_0$$RT0–$$\mathrm{P}_0$$P0, 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 $$\mathrm{RT}_0$$RT0 such that it becomes a modified $$\mathrm{BDM}$$BDM-type element, we develop a new discretization $$\mathrm{BDM}_1^\mathrm{b}$$BDM1b–$$\mathrm{P}_0$$P0. We, therefore, generalize the classical MAC scheme on rectangular grids to triangular grids and retain 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 $$\mathrm{BDM}_1^\mathrm{b}$$BDM1b–$$\mathrm{P}_0$$P0 achieves the optimal convergence rate for both velocity and pressure on general quasi-uniform 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.