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A Stabilized Difference Finite Element Method for the 3D Steady Incompressible Navier-Stokes Equations

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Abstract

In this paper, we develop a stabilized difference finite element (SDFE) method for the 3D steady incompressible Navier-Stokes equations and apply Oseen iterative method to deal with the nonlinear term. Firstly, the finite difference discretization based on finite element pair \(P_1\times P_0\) in the z-direction is used to obtain the finite difference solution \((u^{n}_\tau =\sum ^{l_3}_{k=0}u^{nk}(x,y)\phi _k(z), p^{n}_\tau =\sum ^{l_3}_{k=1}p^{nk}(x,y)\psi _k(z))\), where \((u^{nk}, p^{nk})\) is the solution of 2D linearized Navier-Stokes equations, and then the stabilized finite element discretization based on finite element pair \((P_1,P_1,P_1)\times P_1\) in the (x, y)-plane is used to approximate \((u^{nk}, p^{nk})\), so as to obtain the SDFE solution \((u^{n}_h=\sum ^{l_3}_{k=0}u_{h}^{nk}(x,y)\phi _k(z), p^{n}_h=\sum ^{l_3}_{k=1}p_{h}^{nk}(x,y)\psi _k(z))\) of the 3D linearized Navier-Stokes equations. Our method has the following features. First, difference finite element method overcomes the difficulty of the 3D space discretization. Second, the stabilized method does not require specification of mesh-dependent parameters and retain the symmetry of the original equations. The rigorous stability analysis and error estimate are developed, showing that SDFE method is stable and has optimal convergence. Several numerical tests are presented, confirming the theoretical predictions and verifying the accuracy of the considered method.

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Data Availability

All data generated or analysed during this study are included in this published article [and its supplementary information files].

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Acknowledgements

The authors would like to thank the editor and reviewers for their valuable comments and suggestions that greatly contributed to improving the quality of the present manuscript.

Funding

The work of Pengzhan Huang was supported by Natural Science Foundation (NSF) of China (No.11861067). The work of Xinlong Feng was supported by the Research Fund from Key Laboratory of Xinjiang Province, China (No. 2020D04002), the Scientific Research Plan of Universities in Xinjiang, China (No. XJEDU2020Y001, No.XJEDU2020I001) and the NSF of China (No.U19A2079, No.12071406). The work of Yinnian He was supported by the NSF of China (No. 11771348 and 12026257).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Xiaoli Lu & Yinnian He

  2. College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Pengzhan Huang & Xinlong Feng

Authors
  1. Xiaoli Lu
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  2. Pengzhan Huang
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  3. Xinlong Feng
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  4. Yinnian He
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Contributions

All authors contributed to the study conception and design. Numerical analysis was performed by Xiaoli Lu and Yinnian He. The numerical simulation part was performed by Xiaoli Lu, Pengzhan Huang and Xinlong Feng. The first draft of the manuscript was written by Xiaoli Lu and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yinnian He.

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Lu, X., Huang, P., Feng, X. et al. A Stabilized Difference Finite Element Method for the 3D Steady Incompressible Navier-Stokes Equations. J Sci Comput 92, 104 (2022). https://doi.org/10.1007/s10915-022-01928-2

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  • DOI: https://doi.org/10.1007/s10915-022-01928-2

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