A decoupled, unconditionally energy-stable and structure-preserving finite element scheme for the incompressible MHD equations with magnetic-electric formulation

In this paper, we propose a decoupled, unconditionally energy stable and structure-preserving finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-electric formulation. The spatial discretization is based on mixed finite element method and finite element exterior calculus, where the hydrodynamic unknowns are approximated by stable finite element pair, the magnetic field and electric field are discretized by H(div)-conforming face element and H(curl)-conforming edge element. The time marching is combining the first-order semi-implicit Euler scheme, a first-order stabilized term and some subtle implicit -explicit treatments for the coupling terms. As a result, the fully discrete scheme only needs to solve some linear sub-problems at each time step and yields an exactly divergence-free magnetic field on the discrete level. In particular, we decouple the computations of the magnetic field and electric field equivalently by utilizing the structure-preserving property. Furthermore, the unique solvability and unconditional stability of the scheme are proved. Numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme.