Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D/3D stationary incompressible magnetohydrodynamics

2015 ◽  
Vol 59 (3) ◽  
pp. 589-608 ◽  
Author(s):  
XiaoJing Dong ◽  
YinNian He
Keyword(s):  
Finite Element ◽  
Iterative Methods ◽  
Reynolds Numbers ◽  
Stationary Incompressible Magnetohydrodynamics ◽  
Incompressible Magnetohydrodynamics

The Oseen Type Finite Element Iterative Method for the Stationary Incompressible Magnetohydrodynamics

2017 ◽  
Vol 9 (4) ◽  
pp. 775-794 ◽  
Author(s):  
Xiaojing Dong ◽  
Yinnian He
Keyword(s):  
Reynolds Number ◽  
Finite Element ◽  
Iterative Method ◽  
Mesh Size ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Stability And Convergence ◽  
Stationary Incompressible Magnetohydrodynamics ◽  
Image Position ◽  
Incompressible Magnetohydrodynamics

AbstractIn 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(C0, C1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that . Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Dongyang Shi ◽  
Zhiyun Yu
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Approximate Solutions ◽  
Mixed Finite Element Methods ◽  
Optimal Error Estimates ◽  
Stationary Incompressible Magnetohydrodynamics ◽  
Friedrichs Inequality ◽  
Incompressible Magnetohydrodynamics ◽  
Low Order

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 inL2-norm are established, as well as those in a brokenH1-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.

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