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On Two-Level Oseen Penalty Iteration Methods for the 2D/3D Stationary Incompressible Magnetohydronamics

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Abstract

This paper studies several decoupled penalty methods to overcome the saddle point system of the steady state 2D/3D incompressible magnetohydronamics (MHD). These approaches combine the Oseen iteration and two-level technique with strong uniqueness condition \(0<\frac{\sqrt{2}C_{0}^{2}\max \{1,\sqrt{2}S_{c}\}\Vert {\mathbf{F }}\Vert _{-1}}{(\min \{R_{e}^{-1},S_{c}C_{1}R_{m}^{-1}\})^2}\le 1-\left( \frac{\Vert \mathbf{F }\Vert |_{-1}}{\Vert |\mathbf{F }\Vert _{0}}\right) ^{\frac{1}{2}}<1\) satisfied. For the convenience of implementation, we employ two different simple Lagrange finite element pairs \(P_{1}b-P_{1}-P_{1}b\) and \(P_{1}-P_{0}-P_{1}\) for velocity field, pressure and magnetic field, respectively. Rigorous analysis of the optimal error estimate and stability are provided. We present comprehensive numerical experiments, which indicate the effectiveness of the proposed methods for both two dimensional and three-dimensional problems.

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Acknowledgements

This work is in part supported by the NSF of China (Grant Nos. 11701493) and the GRF of Hong Kong (Grant Nos. 9041980, 9042081)” shoud be replaced by “This work is in part supported by the NSF of Xinjiang Uygur Autonomous Region (Nos. XJEDU2020I001, 2016D01C073, 2019D01C047), the NSF of China (Nos. 11701493, 61962056)”, Tianshan Youth Project of Xinjiang Uygur Autonomous Region (No.2017Q079), 2019 Autonomous Region University Research Program (No.XJEDU2019Y002), the Xinjiang Provincial University Research Foundation of China (No.XJEDU2018I002) and the Matching Projects for Study Abroad of People’s Goverment of Xinjiang Uygur Autonomous Region (No. 2219-51160000313).

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  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Haiyan Su, Xinlong Feng & Jianping Zhao

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  1. Haiyan Su
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Correspondence to Xinlong Feng.

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This work is in part supported by the NSF of China (Grant Nos. 11701493) and the GRF of Hong Kong (Grant Nos. 9041980, 9042081).

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Su, H., Feng, X. & Zhao, J. On Two-Level Oseen Penalty Iteration Methods for the 2D/3D Stationary Incompressible Magnetohydronamics. J Sci Comput 83, 11 (2020). https://doi.org/10.1007/s10915-020-01186-0

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  • DOI: https://doi.org/10.1007/s10915-020-01186-0

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