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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 3/2015

01.06.2015

Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

verfasst von: Long Chen, Ming Wang, Lin Zhong

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2015

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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.
Vorheriger Artikel Pressure Recovery for Weakly Over-Penalized Discontinuous Galerkin Methods for the Stokes Problem
Nächster Artikel A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction–Diffusion Equations on Surfaces

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Metadaten
Titel
Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations
verfasst von
Long Chen
Ming Wang
Lin Zhong
Publikationsdatum
01.06.2015
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2015
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-014-9916-z

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