Pressure and convection robust Finite Elements for Magnetohydrodynamics

We propose and analyze two convection quasi-robust and pressure robust finite element methods for a fully nonlinear time-dependent magnetohydrodynamics problem. Both methods employ the $H_{\rm div}$ conforming BDM element coupled with an appropriate pressure space guaranteeing the exact diagram for the fluid part, and the $H^1$ conforming Lagrange element for the approximation of the magnetic fluxes, and make use of suitable DG upwind terms and CIP stabilizations to handle the fluid and magnetic convective terms. The main difference between the two approaches here proposed (labeled as three-field scheme and four field-scheme respectively) lies in the strategy adopted to enforce the divergence-free condition of the magnetic field. The three-filed scheme implements a grad-div stabilization, whereas the four-field scheme introduces a suitable Lagrange multiplier and additional stabilization terms in the formulation. The developed error estimates for the two schemes are uniform in both diffusion parameters and optimal with respect to the diffusive norm. Furthermore, in the convection dominated regime, being $k$ the degree of the method and $h$ the mesh size, we are able to prove $O(h^k)$ and $O(h^{k+1/2})$ pre-asymptotic error reduction rate for the three-field scheme and four-filed scheme respectively. A set of numerical tests support our theoretical findings.