Volume 9, Issue 4
The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics

Xiaojing Dong & Yinnian He

Adv. Appl. Math. Mech., 9 (2017), pp. 775-794.

Published online: 2018-05

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  • Abstract

In this article, by applying the Stokes projection and an orthogonal projection with respect to curl and div operators, some new error estimates of finite element method (FEM) for the stationary incompressible magnetohydrodynamics (MHD) are obtained. To our knowledge, it is the first time to establish the error bounds which are explicitly dependent on the Reynolds numbers, coupling number and mesh size. On the other hand, The uniform stability and convergence of an Oseen type finite element iterative method for MHD with respect to high hydrodynamic Reynolds number Re and magnetic Reynolds number Rm, or small δ=1−σ with σ= √ 2C 2 0max{1,√ 2Sc}kFk−1/(min{R −1 e ,ScC1R −1 m }) 2 (C0, C1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that h≤(kFk−1/kFk0) 1/2δ. Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.

  • Keywords

Uniform stability, convergence, Oseen type iterative method, finite element method, stationary incompressible magnetohydrodynamics.

  • AMS Subject Headings

35Q30, 65M60, 65N30, 76D05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-775, author = {}, title = {The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {4}, pages = {775--794}, abstract = {

In this article, by applying the Stokes projection and an orthogonal projection with respect to curl and div operators, some new error estimates of finite element method (FEM) for the stationary incompressible magnetohydrodynamics (MHD) are obtained. To our knowledge, it is the first time to establish the error bounds which are explicitly dependent on the Reynolds numbers, coupling number and mesh size. On the other hand, The uniform stability and convergence of an Oseen type finite element iterative method for MHD with respect to high hydrodynamic Reynolds number Re and magnetic Reynolds number Rm, or small δ=1−σ with σ= √ 2C 2 0max{1,√ 2Sc}kFk−1/(min{R −1 e ,ScC1R −1 m }) 2 (C0, C1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that h≤(kFk−1/kFk0) 1/2δ. Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m934}, url = {http://global-sci.org/intro/article_detail/aamm/12175.html} }
TY - JOUR T1 - The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 775 EP - 794 PY - 2018 DA - 2018/05 SN - 9 DO - http://dor.org/10.4208/aamm.2015.m934 UR - https://global-sci.org/intro/aamm/12175.html KW - Uniform stability, convergence, Oseen type iterative method, finite element method, stationary incompressible magnetohydrodynamics. AB -

In this article, by applying the Stokes projection and an orthogonal projection with respect to curl and div operators, some new error estimates of finite element method (FEM) for the stationary incompressible magnetohydrodynamics (MHD) are obtained. To our knowledge, it is the first time to establish the error bounds which are explicitly dependent on the Reynolds numbers, coupling number and mesh size. On the other hand, The uniform stability and convergence of an Oseen type finite element iterative method for MHD with respect to high hydrodynamic Reynolds number Re and magnetic Reynolds number Rm, or small δ=1−σ with σ= √ 2C 2 0max{1,√ 2Sc}kFk−1/(min{R −1 e ,ScC1R −1 m }) 2 (C0, C1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that h≤(kFk−1/kFk0) 1/2δ. Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.

Xiaojing Dong & Yinnian He. (2020). The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics. Advances in Applied Mathematics and Mechanics. 9 (4). 775-794. doi:10.4208/aamm.2015.m934
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