The Arrow–Hurwicz iterative finite element method for the stationary magnetohydrodynamics flow

Yang, Yunbo; Jiang, Yao‐Lin; Kong, Qiongxiang

doi:10.1016/j.amc.2018.10.050

Cited by 4 publications

(11 citation statements)

References 24 publications

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“…The analysis of the optimal error estimate and stability for three methods are provided rigorously. Numerical experiments are in agreement with the theoretical analysis and demonstrate the efficiency of the proposed methods, which indicate that our methods can save much computational cost than the A-H method in [40].…”

Section: Introductionsupporting

confidence: 72%

“…The A-H method and its theoretical analysis are the basis of the two-grid methods, which are very important in the later theoretical analysis of the two-grid A-H methods. Here, we repeat the relevant results from [40] without proofs. Let ρ and α be two positive parameters, we present the A-H method proposed in [40] as follows:…”

Section: The A-h Methodsmentioning

confidence: 89%

“…Here, we repeat the relevant results from [40] without proofs. Let ρ and α be two positive parameters, we present the A-H method proposed in [40] as follows:…”

Section: The A-h Methodsmentioning

confidence: 89%

“…Du and Huang [11] have constructed the A-H method to solve some specific nonlinear saddle-point problems, including the variational multiscale method and the defect-correction method for numerically solving the steady incompressible NS equations. Moreover, the Arrow-Hurwicz iteration for solving the stationary MHD equations has been derived by Yang et al [40] and for solving the stationary thermally coupled incompressible MHD equations has been derived by Keram and Huang [25].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, inspired by [5,12,40], we will study the the two-grid A-H iterative finite element methods to solve the the stationary MHD equations. This paper can be viewed in the framework of Du et al [12] which consider the two-grid A-H methods for solving the NS equations.…”

Section: Introductionmentioning

confidence: 99%

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WITHDRAWN: Two-grid Arrow-Hurwicz iterative finite element methods for the stationary magnetohydrodynamics flow

Yang

Xia

2022

Preprint

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In this paper, we propose and analyze three kinds of two-grid Arrow-Hurwicz (A-H) iterative finite element methodsfor the stationary incompressible magnetohydrodynamics (MHD) equations, which adopt the existingA-H iterative method to obtain the coarse mesh solution, and then correct the solution by threedifferent one-step schemes (Oseen type, Stokes type and Newton type) on the fine mesh. These methods combine the A-H iterative method and the two-grid strategy, maintaining the advantage of both methods and overcoming some of their limitations. Rigorous analysis of the optimal error estimate and stability for three methods are provided. Ample numerical experiments are reported to validate the theoretical results and the efficiency of the numerical schemes.Mathematics Subject Classification (2000) 65N15 · 65N30 · 65N12 · 65M12

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Section: Introductionsupporting

confidence: 72%

Section: The A-h Methodsmentioning

confidence: 89%

“…Here, we repeat the relevant results from [40] without proofs. Let ρ and α be two positive parameters, we present the A-H method proposed in [40] as follows:…”

Section: The A-h Methodsmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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WITHDRAWN: Two-grid Arrow-Hurwicz iterative finite element methods for the stationary magnetohydrodynamics flow

Yang

Xia

2022

Preprint

Self Cite

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Optimal error estimates of two‐level iterative finite element methods for the thermally coupled incompressible MHD with different viscosities

Zhang

2022

Math Methods in App Sciences

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In this paper, we develop and design the two‐level finite element iterative methods for the stationary thermally coupled incompressible MHD equations. The considered numerical schemes are based on the classical iterations with one correction. First, under some strong uniqueness conditions, the rough solutions are obtained on the coarse mesh with the mesh size and the Simple, Oseen, and Newton iterations, respectively. Three kinds of corrections are made by solving a linear problem on the fine mesh with mesh size with different viscosities. Finally, under the weak uniqueness condition, the stationary thermally coupled incompressible MHD is solved by the one‐level finite element Oseen iteration on the fine mesh. The uniform stability and convergence of these two‐level iterative methods are analyzed with respect to the mesh sizes and iterative times . Extensive numerical results are presented to demonstrate the established theoretical findings and show the performances of these two‐level iterative schemes with different viscosities.

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Two‐level methods based on the Arrow–Hurwicz iteration for the steady incompressible magnetohydrodynamic system

Huang

Mahbub

et al. 2023

Numerical Methods Partial

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We consider the two-level methods based on Arrow-Hurwicz (A-H) iteration for solving the stationary incompressible magnetohydrodynamics problem. The methods carry out some A-H iterations on the coarse mesh, improve the obtained solutions by using one-step scheme on the fine mesh. Such methods enjoy some advantages including that the number of A-H iterations can be flexibly and controllably selected, and they are valid for high hydrodynamic Reynolds number. The convergence analyses are established and various numerical experiments are performed to validate and illustrate the stability, accuracy, and efficiency of the new methods.

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The Arrow–Hurwicz iterative finite element method for the stationary magnetohydrodynamics flow

Cited by 4 publications

References 24 publications

WITHDRAWN: Two-grid Arrow-Hurwicz iterative finite element methods for the stationary magnetohydrodynamics flow

WITHDRAWN: Two-grid Arrow-Hurwicz iterative finite element methods for the stationary magnetohydrodynamics flow

Optimal error estimates of two‐level iterative finite element methods for the thermally coupled incompressible MHD with different viscosities

Two‐level methods based on the Arrow–Hurwicz iteration for the steady incompressible magnetohydrodynamic system

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