New analysis of mixed FEMs for dynamical incompressible magnetohydrodynamics

This paper focuses on a new error analysis and a recovering technique of frequently-used mixed FEMs for a dynamical incompressible magnetohydrodynamics (MHD) system. The methods use the standard inf-sup stable Taylor–Hood/MINI velocity-pressure space pairs to solve the Navier–Stokes equations and the Nédélec’s edge element for solving the magnetic field. We establish new and optimal error estimates. In particular, we prove that the method provides the optimal accuracy for the MINI element in L²-norm and for the Taylor-Hood element in H¹-norm. The analysis is based on a modified Maxwell projection and the corresponding estimates in negative norms, while all the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec’s edge approximation in analysis. In addition, at any given time step, we develop a simple recovery technique for numerical approximation to the magnetic field of one order higher accuracy in the spatial direction.