S.G. Janssens said:
By "analytic" do you mean "classical", as in: satisfying the differential form of the PDE?
A classical solution need not exist, and weak solutions may admit discontinuities. In that case, a weak solution would be very far from "differentiable", and the answer to your question would be "yes".
On the other hand, if a classical solution exists, then it is also a weak solution. If, moreover, weak solutions are unique, then the classical solution and the weak solution are the same, and course the answer to your question would be "no".
(This all assumes that the software accurately reproduces the weak solution, including its possible discontinuities.)