Published

Convergence Analysis of The Triangular MAC Scheme for Stokes Equations

Long Chen, Ming Wang, and Lin Zhong

Journal of Scientific Computing, 63(3), 716 - 744, 2015

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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.