Weak solutions and maximal regularity for abstract evolution inequalities

We study the regularity and the approximation of the solution of a parabolic evolution inequality
in the framework of a Hilbert triple.
We give a weak formulation which allows for data with weak regularity and we obtain new existence and regularity results.
We also prove an optimal error estimate for the backward Euler discretization and we apply these results to the porous medium and the Stefan problem.