Some Uzawa-type finite element iterative methods for the steady incompressible magnetohydrodynamic equations

2017 ◽  
Vol 302 ◽  
pp. 34-47 ◽  
Author(s):  
Tielei Zhu ◽  
Haiyan Su ◽  
Xinlong Feng
Keyword(s):  
Finite Element ◽  
Iterative Methods ◽  
Magnetohydrodynamic Equations ◽  
Incompressible Magnetohydrodynamic Equations

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.

Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations

2019 ◽  
Vol 29 (8) ◽  
pp. 2709-2727 ◽  
Author(s):  
Yuan Ping ◽  
Haiyan Su ◽  
Xinlong Feng
Keyword(s):  
Finite Element ◽  
Initial Approximation ◽  
Computational Time ◽  
Magnetohydrodynamic Equations ◽  
Content Type ◽  
Parallel Finite Element ◽  
Incompressible Magnetohydrodynamic Equations ◽  
Element Algorithm ◽  
Element Pair ◽  
Order Element

Purpose The purpose of this paper is to propose a local parallel finite element algorithm based on fully overlapping domain decomposition technique to solve the incompressible magnetohydrodynamic equations. Design/methodology/approach The algorithm uses a lower-order element pair to compute an initial approximation by the Oseen-type iteration and uses a higher-order element pair to solve a linear system in each processor. Findings Besides, the convergence analysis of local parallel finite element algorithm is given. Finally, numerical experiments are presented to verify the efficiency of the proposed algorithm. Originality/value Compared with the numerical solution of the common two-step method, this method is easy to realize and can produce a more accurate solution. And, this approach is executed in parallel, so it saves a lot of computational time.

Exact nonlinear wave solutions of the incompressible magnetohydrodynamic equations in a sheared magnetic field

1990 ◽  
Vol 44 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Hiromitsu Hamabata
Keyword(s):  
Magnetic Field ◽  
Incompressible Fluid ◽  
Large Amplitude ◽  
Uniform Magnetic Field ◽  
Magnetohydrodynamic Equations ◽  
Wave Solutions ◽  
Field Lines ◽  
Incompressible Magnetohydrodynamic Equations ◽  
Physical Quantities ◽  
Nonlinear Wave Solutions

Exact wave solutions of the nonlinear jnagnetohydrodynamic equations for a highly conducting incompressible fluid are obtained for the cases where the physical quantities are independent of one Cartesian co-ordina.te and for where they vary three-dimensionally but both the streamlines and magnetic field lines lie in parallel planes. It is shown that there is a class of exact wave solutions with large amplitude propagating in a straight but non-uniform magnetic field with constant or non-uniform velocity.

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