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Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations.

Long Chen¹, Ming Wang², Lin Zhong¹

¹Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA.

Journal of Scientific Computing

|June 5, 2015

Summary

This study optimizes Stokes equations discretization using a novel finite element method. The new approach achieves optimal convergence rates for velocity and pressure, improving accuracy in fluid dynamics simulations.

Keywords:

Exact divergence free H(div) element Stokes equations

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Area of Science:

Computational fluid dynamics
Numerical analysis
Finite element methods

Background:

Stokes equations are fundamental in fluid dynamics.
Existing discretizations like RT0-P0 for Stokes equations have suboptimal error estimates.
The MAC scheme offers desirable properties but is limited to rectangular grids.

Purpose of the Study:

To develop a new, optimal discretization for the Stokes equations using H(div) elements.
To generalize the MAC scheme to triangular grids while preserving its advantages.
To achieve optimal convergence rates for velocity and pressure.

Main Methods:

Utilizing a modified BDM-type element derived from RT0.
Developing a new discretization scheme for triangular grids.
Applying symmetry and superconvergence results for error analysis.

Main Results:

The proposed discretization achieves optimal convergence rates for velocity and pressure on quasi-uniform grids.
A one and a half order convergence rate is obtained for vorticity and recovered pressure.
The new scheme generalizes the MAC scheme to triangular grids, maintaining exact divergence-free properties and local conservation.

Conclusions:

The novel discretization offers a significant improvement over existing methods for Stokes equations.
The method is applicable to general quasi-uniform grids and retains desirable MAC scheme properties.
Numerical experiments validate the theoretical findings and demonstrate the effectiveness of the proposed scheme.