In an indirect method, my understanding is we convert the Continuous Optimal Control problem to a 2-point boundary problem by using initial conditions on the states and terminal conditions on the costates. 

Then there are the direct methods based on input and/or control discretization. In direct collocation, both states and controls are discretized and a single NonLinear Program is solved.

3 questions then:

1. Is the control trajectory found through indirect/direct approaches is a locally optimal one? (I think it is using the maximum principle so I don't see why this can be a global optimal)
2. How exactly is direct collocation is different then discretizing states/controls/time?
3. Lasty, why is indirect apprach is also called "first optimize, then discretize", where is the discretization part?