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A fully divergence-free finite element method for magneto-hydrodynamic equations

by R. Hiptmair and L.-X. Li and S.-P. Mao and W.-Y. Zheng

(Report number 2017-25)

Abstract
In this paper, a fully divergence-free finite element method is proposed to solve three-dimensional incompressible magnetohydrodynamic equations. The main merit of the method is that the spatial discretization ensures that the approximations of the velocity and the magnetic induction are divergence-free exactly. 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 system of equations to be solved in every timestep. To solve them, we propose an efficient preconditioner so that the number of preconditioned GMRES iterations is uniform with respect to the number of degrees of freedom. Moreover, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.

Keywords: Magnetohydrodynamic equations; divergence-free finite element method; preconditioner; magnetic vector potential; driven cavity flow.

@Techreport{HLMZ17_721,
  author = {R. Hiptmair and L.-X. Li and S.-P. Mao and W.-Y. Zheng},
  title = {A fully divergence-free finite element method for magneto-hydrodynamic equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-25.pdf },
  year = {2017}
}

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