  We propose and analyze two convection quasi-robust and pressure robust finite
element methods for a fully nonlinear time-dependent magnetohydrodynamics
problem. The schemes make use of suitable upwind and CIP stabilizations to
handle the fluid and magnetic convective terms. The developed error estimates
are uniform in both diffusion parameters and optimal with respect to the
diffusive norm; furthermore, for the second (more complex) method we are able
to show a quicker error reduction rate in convection dominated regimes. A set
of numerical tests support our theoretical findings.
