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Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations

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Abstract

The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.

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Funding

This work is partly supported by the NSF of China (No. 11701493, 12061076, 12061075), the NSF of Xinjiang Province (No. 2016D01C073), Tianshan Youth Project of Xinjiang Province (No. 2017Q079), and Z. Li is partially supported by a Simons grant 633724.

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Authors and Affiliations

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

    Jiali Xu & Haiyan Su

  2. CRSC & Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Zhilin Li

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  1. Jiali Xu
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  2. Haiyan Su
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  3. Zhilin Li
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Correspondence to Haiyan Su.

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Xu, J., Su, H. & Li, Z. Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations. Numer Algor 90, 1117–1151 (2022). https://doi.org/10.1007/s11075-021-01224-4

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