Is it possible to improve the efficiency of solving a non-relativistic problem by using a relativistic theory? There are many beautiful theories, especially quantum field theory, start with covariant formulation compatible with relativity. But apparently most phenomena in atomic physics, material sciences, chemistry, biology, etc., are well-governed under non-relativistic physical rules. Is it possible, or is there any case already existed, that a relativistic or relativity-like formulation can simplify a non-relativistic problem?
 A: A good example would be the problem of cosmological evolution of homogeneous and isotropic universe filled with gravitating matter.
When considered within the framework of general relativity theory the solution to the problem is the Friedmann–Lemaître–Robertson–Walker metric, a relatively simple spacetime determined by  two ordinary differential equations that has been found within years after publication of general relativity theory and is described in almost every GR textbook.
That same problem, when considered within the framework of Newtonian gravitational theory (referred to as “Newtonian cosmology”) has not been solved satisfactorily until decades after the relativistic case, though originally it was formulated by Newton himself who gave an erroneous solution (that such universe would be static).
The difficulties of the Newtonian case are traceable to the fact that infinite number of bodies attracting one another  via Newton's law produce diverging results (if done in the straightforward way), whereas relativistic Einstein's field equation describe how local spacetime geometry is determined by local matter content. Once a suitable set of local variables is introduced in the Newtonian problem, it becomes solvable as easily as its relativistic counterpart.
For historical account and discussion of conceptual problems in Newtonian cosmology see the paper by Norton:

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*Norton, J. D. (1999). The cosmological woes of Newtonian gravitation theory. The expanding worlds of General Relativity: Einstein studies, 7, 271-323, pdf on author's homepage.

While for pedagogical exposition of the actual rigorous solution for that problem see Tipler:

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*Tipler, F. J. (1996). Rigorous Newtonian cosmology. American Journal of Physics, 64(10), 1311-1315, doi:10.1119/1.18398.

