for the same geometry and material property, analytic solution will not differ, even for extensive mesh refinement. it will always be the same. the analytic solution is derived from solid mechanics and there's no function of mesh or element.
from http://docs.sfepy.org/doc/primer.html#problem-statement -

The tensile and compressive stresses that develop in the specimen as a result of the point loads P are a function of the diameter (D) and thickness (t) of the cylindrical specimen. At the centre of the specimen, the compressive stress is 3 times the tensile stress and the analytical formulation for these are, respectively:

(1)\sigma_t=\frac{2P}{\pi tD}

(2)\sigma_c=\frac{6P}{\pi tD}