  We propose consistent locally stabilized, conforming finite element schemes
on completely unstructured simplicial space-time meshes for the numerical
solution of non-autonomous parabolic evolution problems under the assumption of
maximal parabolic regularity. We present new a priori estimates for
low-regularity solutions. In order to avoid reduced convergence rates appearing
in the case of uniform mesh refinement, we also consider adaptive refinement
procedures based on residual a posteriori error indicators. The huge system of
space-time finite element equations is then solved by means of GMRES
preconditioned by algebraic multigrid.
