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Two-level iterative finite element methods for the stationary natural convection equations with different viscosities based on three corrections

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Abstract

This paper considers the two-level iterative finite element methods for the steady natural convection equations under some uniqueness conditions with the Simple-, Oseen- and Newton-type corrections. Firstly, the stability and convergence of the one-level iterative finite element methods are analyzed under some restrictions on physical parameters. Secondly, under the strong uniqueness condition, we develop the two-level finite element method with Simple, Oseen and Newton iterations of m times on the coarse mesh \(\tau _H\) with mesh size H, and then, the considered problem is linearized in three correction schemes with the Simple, Oseen and Newton corrections one time on the fine grid \(\tau _h\) with mesh size \(h\ll H\) based on the obtained iterative solutions. From the theoretical point of view, the results obtained by the two-level iterative methods have the same precision as those obtained by the one-level method which mesh sizes satisfy \(h={\mathcal {O}}(H^2)\) and the iterative steps are greater than some constants. Thirdly, the stability and convergence of one-level Oseen iterative scheme with respect to the mesh size and the iterative time m are provided under a weak uniqueness condition. Finally, some numerical experiments are designed to confirm the established theoretical findings and verify the performance of the proposed numerical schemes.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, People’s Republic of China

    Huifang Zhang, Chuanjun Chen & Tong Zhang

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  1. Huifang Zhang
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  2. Chuanjun Chen
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  3. Tong Zhang
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Correspondence to Tong Zhang.

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Communicated by Frederic Valentin.

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This work was supported by National Natural Science Foundation of China (nos. 11971152, 12271468), Shandong Province Natural Science Foundation (nos. ZR2021ZD03, ZR2021MA010 ) and the Natural Science Foundation of Henan Province (no. 202300410167)

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Zhang, H., Chen, C. & Zhang, T. Two-level iterative finite element methods for the stationary natural convection equations with different viscosities based on three corrections. Comp. Appl. Math. 42, 11 (2023). https://doi.org/10.1007/s40314-022-02147-z

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