This talk focusses on the smoothness of the solutions of parabolic PDEs on Lipschitz domains in the fractional Sobolev scale H^s, s in R.

The regularity in these spaces is related with the approximation order that can be achieved by numerical schemes based on uniform grid refinements.

The results presented provide a first attempt to generalize the well-known H^3/2-Theorem of Jerison and Kenig to parabolic PDEs. As a special case the heat equation on  radial-symmetric cones is investigated.