I have had this sme observation, similar toRon's comment. Refining the element refines the displacement. You can see this from Hooke's Law, F=kd, k=stiffness matrix. Since F=P\A, stress over area, then changing the stiffness matrix also influences displacement and stress.
Changing means refining, in the sense that stable models should exponentially move towards a limit. This is one of the checks for stability, a small change in element size should not yield wild swings in stress nor displacement. What you are doing is akin to adjusting an answer based upon significant digits.
But this should be a question in the FEA forum. You'd probably get better discussions.
Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada