A fully divergence-free finite element method for magnetohydrodynamic equations

Abstract

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.

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

“…In recent years, the motion of an electrically conducting fluid within a three‐dimensional lid‐driven cavity has been extensively investigated in the literature . In here, the calculations are carried out for a lid‐driven cubic cavity with a horizontally applied magnetic field, B =(1,0,0) ⊤ corresponding to the work of Li and Zheng due to its relatively high Hartmann number.…”

A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations

“…Sermane and Temam proved the existence and uniqueness of local strong solution on regular domains [33]. There have been numerous works on numerical methods for the incompressible MHD equations, see [3,14,17,21,22,25,29,31,35]. Due to the nonlinear coupling of the unknowns, the divergence free constraints and the low regularity of the exact solutions, it is a challenging task to design efficient numerical schemes for the dynamical incompressible MHD equations.…”

A Convergent Linearized Lagrange Finite Element Method for the Magneto-hydrodynamic Equations in Two-Dimensional Nonsmooth and Nonconvex Domains

“…• As far as we know, the only existing proofs for the (subsequence) convergence of numerical solutions in possibly nonsmooth domains were given in [35] and [25]. In [35], Prohl studied several fully discrete linearized FEMs (with different time discretization and decoupling methods) with curl-conforming Nédélec edge elements for the magnetic field, and proved the convergence of two numerical schemes to weak solutions under the mesh restrictions τ = O(h 4 ) and τ = O(h 3 ), respectively, where τ denotes the time-step size and h the spacial mesh size.…”

“…Without such mesh restrictions, the weak * convergence of numerical solutions in L ∞ (0, T ; L 2 (Ω)) was proved. In [25], Hiptmair, Li, Mao and Zheng discretized a magnetic potential formulation of the three-dimensional MHD equations:…”

“…These methods focus on approximating the magnetic field H in the Maxwell equations directly by using H 1 -conforming finite element methods. Similarly as [25,35], the proof of convergence in this paper is only based on the regularity of the initial conditions and source terms, without any extra assumptions on the regularity of the solution. Strong convergence of subsequences in L 2 (0, T ; L 2 (Ω)) as (τ n , h n ) → (0, 0) is proved for the numerical solutions of u and H without mesh restrictions.…”