Journal of Applied Mathematics
Volume 2012 (2012), Article ID 825609, 21 pages
http://dx.doi.org/10.1155/2012/825609
Research Article

Low-Order Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics Equations

Dongyang Shi1 and Zhiyun Yu1,2

1Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China
2College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

Received 14 February 2012; Accepted 2 April 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 Dongyang Shi and Zhiyun Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The nonconforming mixed finite element methods (NMFEMs) are introduced and analyzed for the numerical discretization of a nonlinear, fully coupled stationary incompressible magnetohydrodynamics (MHD) problem in 3D. A family of the low-order elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field, and the magnetic field. The existence and uniqueness of the approximate solutions are shown, and the optimal error estimates for the corresponding unknown variables in 𝐿 2 -norm are established, as well as those in a broken 𝐻 1 -norm for the velocity and the magnetic fields. Furthermore, a new approach is adopted to prove the discrete Poincaré-Friedrichs inequality, which is easier than that of the previous literature.