Structure-preserving Finite Element Methods for Stationary MHD Models
It provides structure-preserving numerical methods for MHD simulations, ensuring physical constraints are exactly satisfied, which is important for accurate long-time simulations in plasma physics and engineering.
This paper develops mixed finite element schemes for stationary MHD models that exactly preserve Gauss's law and the energy law, proving well-posedness, existence of solutions, and convergence of Picard iterations and finite element methods.
In this paper, we develop a class of mixed finite element scheme for stationary magnetohydrodynamics (MHD) models, using magnetic field $\bm B$ and current density $\bm j$ as the discretization variables. We show that the Gauss's law for the magnetic field, namely $\nabla\cdot\bm{B}=0$, and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for $H^{h}(\mathrm{div})$, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of Picard iterations and finite element methods under some conditions.