  In this paper we consider parabolic problems with stress tensor depending
only on the symmetric gradient. By developing a new approximation method (which
allows to use energy-type methods typical for linear problems) we provide an
approach to obtain global regularity results valid for general potential
operators with $(p,\delta)$-structure, for all $p>1$ and for all $\delta>0$. In
this way we prove ``natural'' second order spatial regularity -- up to the
boundary -- in the case of homogeneous Dirichlet boundary conditions. The
regularity results, are presented with full details for the parabolic setting
in the case $p>2$. However, the same method also yields regularity in the
elliptic case and for $1<p\leq 2$, thus proving in a different way results
already known.
