title: Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations.
abstract: In this paper, we consider the use ofH(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 RT0such 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.
authors.affiliation: Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA.
authors.forename: Long
authors.initials: L
authors.lastname: Chen
authors.affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.
authors.forename: Ming
authors.initials: M
authors.lastname: Wang
authors.affiliation: Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA.
authors.forename: Lin
authors.initials: L
authors.lastname: Zhong
keywords: Exact divergence free; H(div) element; Stokes equations