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Volume 10, Issue 4
Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics

D. Shi & Z. Yu

Int. J. Numer. Anal. Mod., 10 (2013), pp. 904-919.

Published online: 2013-10

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  • Abstract

The main aim of this paper is to study the approximation of nonconforming mixed finite element methods for stationary, incompressible magnetohydrodynamics (MHD) equations in 3D. A family of nonconforming finite elements are taken as the approximation spaces for the velocity field, the piecewise constant element for the pressure and the Nédélec's element with the lowest order for the magnetic field on hexahedra or tetrahedra. A new simple method is adopted to prove the discrete Poincaré-Friedrichs inequality instead of the discrete Helmholtz decomposition method. The existence and uniqueness of the approximate solutions are shown. The convergence analysis is presented and the optimal order error estimates for the pressure in $L^2$-norm, as well as those in a broken $H^1$-norm for the velocity field and H($curl$)-norm for the magnetic field are derived.

  • Keywords

Incompressible MHD equations, Nonconforming mixed finite element method, Optimal error estimates.

  • AMS Subject Headings

65N30, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-904, author = {D. Shi and Z. Yu}, title = {Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {4}, pages = {904--919}, abstract = {

The main aim of this paper is to study the approximation of nonconforming mixed finite element methods for stationary, incompressible magnetohydrodynamics (MHD) equations in 3D. A family of nonconforming finite elements are taken as the approximation spaces for the velocity field, the piecewise constant element for the pressure and the Nédélec's element with the lowest order for the magnetic field on hexahedra or tetrahedra. A new simple method is adopted to prove the discrete Poincaré-Friedrichs inequality instead of the discrete Helmholtz decomposition method. The existence and uniqueness of the approximate solutions are shown. The convergence analysis is presented and the optimal order error estimates for the pressure in $L^2$-norm, as well as those in a broken $H^1$-norm for the velocity field and H($curl$)-norm for the magnetic field are derived.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/603.html} }
TY - JOUR T1 - Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics AU - D. Shi & Z. Yu JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 904 EP - 919 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/603.html KW - Incompressible MHD equations, Nonconforming mixed finite element method, Optimal error estimates. AB -

The main aim of this paper is to study the approximation of nonconforming mixed finite element methods for stationary, incompressible magnetohydrodynamics (MHD) equations in 3D. A family of nonconforming finite elements are taken as the approximation spaces for the velocity field, the piecewise constant element for the pressure and the Nédélec's element with the lowest order for the magnetic field on hexahedra or tetrahedra. A new simple method is adopted to prove the discrete Poincaré-Friedrichs inequality instead of the discrete Helmholtz decomposition method. The existence and uniqueness of the approximate solutions are shown. The convergence analysis is presented and the optimal order error estimates for the pressure in $L^2$-norm, as well as those in a broken $H^1$-norm for the velocity field and H($curl$)-norm for the magnetic field are derived.

D. Shi and Z. Yu. (2013). Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics. International Journal of Numerical Analysis and Modeling. 10 (4). 904-919. doi:
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