In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, perhaps allowing the use of unstructured space-time meshes. Although timestepping methods may be easier to implement, is there a particular reason why space-time meshing is not viable? I imagine one has to tailor meshes to respect the physical properties of a given problem, but I am not certain.