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 wellknown H^{3/2}Theorem of Jerison and Kenig to parabolic PDEs. As a special case the heat equation on radialsymmetric cones is investigated.
