Analysis of a semi-implicit and structure-preserving finite element method for the incompressible MHD equations with magnetic-current formulation. (English) Zbl 1531.65188
Summary: In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in [K. Hu and J. Xu, Sci. Sin., Math. 46, No. 7, 967–980 (2016; Zbl 1499.76072)]. We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
Citations:
Zbl 1499.76072Software:
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