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A Parallel Finite Element Discretization Algorithm Based on Grad-Div Stabilization for the Navier–Stokes Equations

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Abstract

We present and study a parallel grad-div stabilized finite element discretization algorithm based on entire-overlapping domain decomposition for the numerical simulation of Navier–Stokes equations. The algorithm is easy to implement on top of existing sequential software, in which each subproblem used to calculate a local solution in its designated subregion is actually a global problem with vast of degrees of freedom coming from its own subregion, and hence, can be solved independently with other subproblems. We derive error bounds of the approximate solution by employing the technical tool of local a priori estimate, and investigate the effect of grad-div stabilization term on the approximation solutions. Numerical comparisons, with both inf-sup stable and unstable mixed finite elements pairs for the velocity and pressure, show that our present algorithm has an amazing superiority to its counterpart without stabilization in the sense that accuracy of the approximate velocities could be improved by two orders of magnitude when the viscosity \(\nu \) is small. While compared with the usual standard serial grad-div stabilized finite element method, our algorithm saves lots of CPU time in computing a solution with comparable accuracy.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to express their deep gratitude to the anonymous referees who made valuable suggestions for revision and improvement of the manuscript.

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  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Yueqiang Shang, Jiali Zhu & Bo Zheng

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  1. Yueqiang Shang
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This work was supported by the Natural Science Foundation of Chongqing Municipality, China (No. cstc2021jcyj-msxmX1044).

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Shang, Y., Zhu, J. & Zheng, B. A Parallel Finite Element Discretization Algorithm Based on Grad-Div Stabilization for the Navier–Stokes Equations. J. Math. Fluid Mech. 26, 42 (2024). https://doi.org/10.1007/s00021-024-00868-1

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