2014/10/04 by Kaibo Hu, Hu, Kaibo, Yicong Ma +3 · 1 citation
Engineering · #65M60 (Primary) #65Z05 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1410.1095
openalex publication_date 2014/10/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper is devoted to the design and analysis of some structure-preserving finite element schemes for the magnetohydrodynamics (MHD) system. The main feature of the method is that it naturally preserves the important Gauss law, namely ∇⋅\boldsymbolB=0. In contrast to most existing approaches that eliminate the electrical field variable \boldsymbolE and give a direct discretization of the magnetic field, our new approach discretizes the electric field \boldsymbolE by Nédélec type edge elements for H(curl), while the magnetic field \boldsymbolB by Raviart-Thomas type face elements for H(div). As a result, the divergence-free condition on the magnetic field holds exactly on the discrete level. For this new finite element method, an energy stability estimate can be naturally established in an analogous way as in the continuous case. Furthermore, well-posedness is rigorously established in the paper for both the Picard and Newton linearization of the fully nonlinear systems by using the Brezzi theory for both the continuous and discrete cases. This well-posedness naturally leads to robust (and optimal) preconditioners for the linearized systems.