Its usually only finite difference methods that are not variational. But finite difference is dominant in academia, not in industry. And that is changing as well with methods such as the discontinuous Galerkin method. The more popular finite volume method in industry, can also be seen as a variational problem.
Yes, I exaggerated when I said that, but its still mostly variational problems.
> The more popular finite volume method in industry, can also be seen as a variational problem.
By that standard, you could interpret almost any numerical method for PDEs used in academia or industry as variational (aside from some fringe ones). By "variational" I mean methods which are designed in a variational way from the start, not can be merely interpreted variationally.
Well, it helps to see these connections. For example, realising that the lowest order DG method is finite volume lets one think about how to extend well studied finite volume properties to high order DG methods.
Not true. In computational fluid dynamics, variational methods are only one category out of many, and they aren't dominant.