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Optimal error estimates of penalty difference finite element method for the 3D steady Navier-Stokes equations

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Abstract

In this paper, a penalty difference finite element (PDFE) method is presented for the 3D steady Navier-Stokes equations by using the finite element space pair \((P_1^b, P_1^b, P_1) \times P_1\) in the direction of (x, y), where the finite element space pair \((P_1^b, P_1^b) \times P_1\) satisfies the discrete inf-sup condition in a 2D domain \(\omega \). This new method consists of transmitting the finite element solution \((u_h,p_h)\) of the 3D steady Navier-Stokes equations in the direction of (x, y, z) into a series of the finite element solution pair \((u_h^{nk},p_h^{nk})\) based on the 2D finite element space pair \((P_1^b, P_1^b, P_1)\times P_1\), which can be solved by the 2D decoupled penalty Oseen iterative equations. Moreover, the PDFE method of the 3D steady Navier-Stokes equations is well designed and the \(H^1-L^2\)-optimal error estimate with respect to \((\varepsilon , \sigma ^{n+1}, h, \tau )\) of the numerical solution \((u^n_h,p_h^n)\) to the exact solution \((\tilde{u},\tilde{p})\) is provided. Here \(0<\varepsilon<<1\) is a penalty parameter, \(\sigma =\frac{N}{\nu ^2}\Vert F\Vert _{-1,\Omega }\) is the uniqueness index, n is a iterative step number, \(\tau \) is a mesh size in the direction of z and h is a mesh size in the direction of (x, y). Finally, numerical tests are presented to show the effectiveness of the PDFE method for the steady Navier-Stokes equations.

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The data sets generated during the current study are available from the authors on reasonable request.

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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 Y. He was supported by the National Science Foundation (NSF) of China (No. 12271465). The work of X. Feng was supported by the NSF of Xinjiang Province (No. 2022TSYCTD0019 and No. 2022D01D32), the NSF of China (12071406) and the Foundation of National Key Laboratory of Computational Physics (No. 6142A05230203).

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  1. Xiaoli Lu and Yinnian He contributed equally to this work.

Authors and Affiliations

  1. College of Mathematics and Systems Science, Xinjiang University, Shengli Road, Urumqi, 830046, Xinjiang, China

    Xinlong Feng & Yinnian He

  2. School of Science, Chang’an University, South Second Ring Road, Xi’an, 710064, Shaanxi, China

    Xiaoli Lu

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xianning West Road, Xi’an, 710049, Shaanxi, China

    Yinnian He

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  1. Xinlong Feng
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Xinlong Feng, Xiaoli Lu and Yinnian He have participated sufficiently in the work to take responsibility for the content, including participation in the concept, method, analysis and writing. All authors certify that this manuscript has not been submitted to other journals for publication.

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Correspondence to Yinnian He.

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Feng, X., Lu, X. & He, Y. Optimal error estimates of penalty difference finite element method for the 3D steady Navier-Stokes equations. Numer Algor 98, 1395–1427 (2025). https://doi.org/10.1007/s11075-024-01838-4

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