As a side note - if you know a little electrostatics, recall that poisson's equation (the PDE div(grad(v))=F describes the electric potential v, where the electric field is the gradient of the potential. E=grad(v). Then also recall that the energy stored in a static electric field is proportional to E^2 = |grad(v)|^2. Thus the term "energy method" is somewhat appropriate.
The same sort of thing applies in other situations described by the same PDE (eg: steady-state heat equation, darcy flow, diffusion, and so on). |grad(v)|^2 will describe an energy. If you have some physics background go ahead and think about some of these situations a bit. It is kind of surprising that it all works out like that for so many widely different physical phenomenon.