What are the biggest problems about the numerical, finite-element GR models? As I know, for example the modelling of the collapse of a neutron star (to a black hole) wasn't done correctly until now.

Why? Yes, I know, the Einstein Field Equations aren't really easy to solve. But, for example, the situation is very similar in the case of the QCD, but there are already fruitful results existing ( http://physicsworld.com/cws/article/news/2008/nov/21/proton-and-neutron-masses-calculated-from-first-principles ).

Is there any theoretical obstacle in the way? Or we simply didn't have enough fast computers/physicist/programmer manyear until now?
If you can provide examples of numerical methods in GR you've seen/heard of that would help focus the question.

From the article you linked to: "The technique keeps track of a vast number of quarks and gluons by describing the space and time inside a proton with a set of points that make up a 4D lattice". This almost gets to the main issue with Numerical Relativity. There is no natural computational grid on which to simulate space-time. The whole game with GR is that gravity is space-time so first you have to simulate the space-time and then you have to simulate the objects (neutron stars, black holes, gravitational waves) on top. 

As the links below go into, its very difficult to create a consistent computational grid since the physical space-time your trying to simulate for a black hole has "funny" things in it like singularities, or an event horizon pas which we can't really know what's going on. 

I think this article: http://astronomy.com/magazine/2016/02/putting-einstein-to-the-test?page=1

does a good job of summing up the field, and its quite accessible.

For something more rigorous please see: http://arxiv.org/pdf/1010.5260v2.pdf
That paper gets into some of the math behind the article linked to above.