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01-06-2015 | Issue 3/2015

Journal of Scientific Computing 3/2015

Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

Journal:
Journal of Scientific Computing > Issue 3/2015
Authors:
Long Chen, Ming Wang, Lin Zhong
Important notes
The authors Long Chen and Lin Zhong was supported by NSF Grant DMS-1115961, and in part by Department of Energy prime award # DE-SC0006903, NSF grant DMS-1161621 and NIH grant P50GM76516. The work of the author Ming Wang was supported by 2010–2012 China Scholarship Council (CSC).

Abstract

In this paper, we consider the use of \(H(\mathrm{div })\) elements in the velocity–pressure formulation to discretize Stokes equations in two dimensions. We address the error estimate of the element pair \(\mathrm{RT}_0\)\(\mathrm{P}_0\), 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 \(\mathrm{RT}_0\) such that it becomes a modified \(\mathrm{BDM}\)-type element, we develop a new discretization \(\mathrm{BDM}_1^\mathrm{b}\)\(\mathrm{P}_0\). 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 \(\mathrm{BDM}_1^\mathrm{b}\)\(\mathrm{P}_0\) 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.

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