  Similar evolutionary variational and quasi-variational inequalities with
gradient constraints arise in the modeling of growing sandpiles and type-II
superconductors. Recently, mixed formulations of these inequalities were used
for establishing existence results in the quasi-variational inequality case.
Such formulations, and this is an additional advantage, made it possible to
determine numerically not only the primal variables, e.g. the evolving sand
surface and the magnetic field for sandpiles and superconductors, respectively,
but also the dual variables, the sand flux and the electric field.
  Numerical approximations of these mixed formulations in previous works
employed the Raviart--Thomas element of the lowest order. Here we introduce
simpler numerical approximations of these mixed formulations based on the
nonconforming linear finite element. We prove (subsequence) convergence of
these approximations, and illustrate their effectiveness by numerical
experiments.
