Model: two parts (x and y) in contact together (tie constraint) under some forces on part x.

Goal: Find the reasonable mesh size combination for two parts. 

I tried to perform a mesh convergence study via decreasing the global seed size. I changed the global seed size of part x from 8mm, to 4mm, to 2mm and then 1mm. Judging from the max displacement output, it seems to converge below x=2. But it won't converge if I use the max stress as the metric. And I noticed a change in the local stress distribution on the contacting surface bewteen part x and y, as shown below. I'm not sure why there is a high local stress within one element. Apart from that bit, everything else looks quite similar. I suspect it's an artifact or maybe overconstrian related to the tie constraint. Should I further change the mesh, for instance, using the adaptive mesh? But that would make the mesh convergence study a bit difficult to keep track of... What's the best practice here?

P.S. if I further reduce the mesh size (x=0.5mm), I get a fatal error: the problem is too large to run with Abaqus.