Hi, i am dealing with the same sort of topic. First let me see if I can make something more clear.

(1) is a PDE (partial differential equation) and (3) is the minimization problem which is derived from (1). Finding a solution for (3) will take some numerical methods. But this solution is also a solution for (1).

Now from (3) to (1):

Let be the function minimizing this

Substitute this in (3) minimizing problem

Now differentiate with respect to , and then E(u) reaches a minimum for . You should get the following:

. I can't derive this but you will get the following;

Use the divergence theorem to get rid of

And use DuBois' Lemma to derive the PDE:

with the boundary condition:

Which is the natural boundary condition. The essential boundary condition was already given, u=0 on Gamma.

I skipped some essential steps cause these are not clear to me either, but I am pretty shure this is the way to go from PDE to minimizing problem.

From minimizing to FEM is the next step. Hope someone can contribute to this!!

Bye Alex