Azkaban Prison

The origin story for Azkaban is interwoven with that for Dementors. It starts out with the idea that these beasts cannot be killed, and are "determined to cling to" the building/island.¹ It finishes with the idea that they have been purged by the first Minister to actually try.

This illogic is unfortunately typical. We do not know if the dementors were in fact destroyed or relocated. We do not know why they were thought to be indestructible, nor how the purge was accomplished. We do not know what was done with the dark magic residue embedded in the walls.