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A three-step defect-correction algorithm for incompressible flows with friction boundary conditions

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Abstract

Based on finite element discretization and a recent variational multiscale-stabilized method, we propose a three-step defect-correction algorithm for solving the stationary incompressible Navier-Stokes equations with large Reynolds numbers, where nonlinear slip boundary conditions of friction type are considered. This proposed algorithm consists of solving one nonlinear Navier-Stokes type variational inequality problem on a coarse grid in a defect step, and solving two stabilized and linearized Navier-Stokes type variational inequality problems which have the same stiffness matrices with only different right-hand sides on a fine grid in correction steps. In the defect step, an artificial viscosity term is used as a stabilizing factor, making the nonlinear system easier to solve. Error bounds of the approximate solutions in L2 norms for the velocity gradient and pressure are estimated. Scalings of the algorithmic parameters are derived. Some numerical results are given to support the theoretical predictions and test the validity of the present algorithm.

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Acknowledgements

The authors would like to express their deep gratitude to the anonymous referees for their valuable comments and suggestions, which led to a large improvement of the manuscript.

Funding

This work was supported by the Natural Science Foundation of China (No. 11361016), the Natural Science Foundation of Chongqing Municipality, China (No. cstc2021jcyj-msxmX1044), and Graduate Scientific Research Innovation Project of Chongqing Municipality, China (No. CYB21095).

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

    Bo Zheng & Yueqiang Shang

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  1. Bo Zheng
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Correspondence to Yueqiang Shang.

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Zheng, B., Shang, Y. A three-step defect-correction algorithm for incompressible flows with friction boundary conditions. Numer Algor 91, 1483–1510 (2022). https://doi.org/10.1007/s11075-022-01311-0

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