[coherentpony]

Approximate solutions are found computationally.

Non-trivial closed-form solutions are hopeless. Existence of solutions is a little less hopeless. Existence of solutions over small time horizons has been proven. This is all in three space dimensions.

There is a theorem stating that if global-time solutions exist and are unique, then computationally computed approximate solutions are 'good'. Moreover, they get better as you refine the computational domain.

[noname123]

Is finite element method good enough? Can you provide a concrete example where approximation is not good enough.

[coherentpony]

Special approximations called 'Galerkin approximations' converge.

Finite element methods are a special case of Galerkin methods. Finite element methods are good enough.