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Volume 41, Issue 4
A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System

Zhiyun Yu, Dongyang Shi & Huiqing Zhu

DOI: 10.4208/jcm.2107-m2021-0114

J. Comp. Math., 41 (2023), pp. 569-587.

Published online: 2023-02

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  • Abstract

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.

  • Keywords

Non-stationary incompressible MHD problem, Nonconforming mixed FEM, Optimal order error estimates.

  • AMS Subject Headings

65N15, 65N30, 65M60, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

5772@zut.edu.cn (Zhiyun Yu)

shi_dy@zzu.edu.cn (Dongyang Shi)

Huiqing.Zhu@usm.edu (Huiqing Zhu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-569, author = {Yu , ZhiyunShi , Dongyang and Zhu , Huiqing}, title = {A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {4}, pages = {569--587}, abstract = {

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2107-m2021-0114}, url = {http://global-sci.org/intro/article_detail/jcm/21406.html} }
TY - JOUR T1 - A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System AU - Yu , Zhiyun AU - Shi , Dongyang AU - Zhu , Huiqing JO - Journal of Computational Mathematics VL - 4 SP - 569 EP - 587 PY - 2023 DA - 2023/02 SN - 41 DO - http://doi.org/10.4208/jcm.2107-m2021-0114 UR - https://global-sci.org/intro/article_detail/jcm/21406.html KW - Non-stationary incompressible MHD problem, Nonconforming mixed FEM, Optimal order error estimates. AB -

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.

Zhiyun Yu, Dongyang Shi & Huiqing Zhu. (2023). A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System. Journal of Computational Mathematics. 41 (4). 569-587. doi:10.4208/jcm.2107-m2021-0114
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