Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

In this paper, we consider the use of H(div) elements in the velocity-pressure formulation to discretize Stokes equations in two dimensions. We address the error estimate of the element pair RT0-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 inter-polant. By enlarging RT0 such that it becomes a modified BDM-type element, we develop a new discretization [Formula: see text]. 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 [Formula: see text] 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.