Decoupled and linearized scalar auxiliary variable finite element method for the time‐dependent incompressible magnetohydrodynamic equations: Unconditional stability and convergence analysis

2021 ◽  
Author(s):  
Tong Zhang ◽  
Jinting Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Analysis ◽  
Auxiliary Variable ◽  
Unconditional Stability ◽  
Time Dependent ◽  
Stability And Convergence ◽  
Magnetohydrodynamic Equations ◽  
Stability And Convergence Analysis ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document