RETRACTED ARTICLE: Global strong solution to the three-dimensional stochastic incompressible magnetohydrodynamic equations

2015 ◽  
Vol 365 (3-4) ◽  
pp. 1219-1256 ◽  
Author(s):  
Zhong Tan ◽  
Dehua Wang ◽  
Huaqiao Wang
Keyword(s):  
Strong Solution ◽  
Three Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Global Strong Solution ◽  
Retracted Article ◽  
Incompressible Magnetohydrodynamic Equations

A LOGARITHMALLY IMPROVED BLOW-UP CRITERION OF SMOOTH SOLUTIONS FOR THE THREE-DIMENSIONAL MHD EQUATIONS

2012 ◽  
Vol 23 (02) ◽  
pp. 1250027 ◽  
Author(s):  
YU-ZHU WANG ◽  
HENGJUN ZHAO ◽  
YIN-XIA WANG
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Three Dimensional ◽  
Mhd Equations ◽  
Magnetohydrodynamic Equations ◽  
Smooth Solutions ◽  
The Cauchy Problem ◽  
Incompressible Magnetohydrodynamic Equations ◽  
Blow Up Criterion

In this paper we investigate the Cauchy problem for the three-dimensional incompressible magnetohydrodynamic equations. A logarithmal improved blow-up criterion of smooth solutions is obtained.

A fully divergence-free finite element method for magnetohydrodynamic equations

2018 ◽  
Vol 28 (04) ◽  
pp. 659-695 ◽  
Author(s):  
Ralf Hiptmair ◽  
Lingxiao Li ◽  
Shipeng Mao ◽  
Weiying Zheng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Divergence Free ◽  
Incompressible Magnetohydrodynamic Equations ◽  
Preconditioned Gmres ◽  
Energy Law ◽  
Element Method

We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom. As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.

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