I've seen it quoted on internet sites that Augustine said that the same fire that tortures the damned purifies the elect. Supposedly he meant by this that the fires of Purgatory and Hell are the same. Where does this quote come from?
This quote does come from Augustine, but its application to the doctrine of purgatory seems to be the work of Thomas Aquinas. Aquinas (who mistakenly attributes it to Gregory the Great) quotes similar language in his second article on purgatory from the appendices of the Summa, saying:
On the contrary, Gregory says, "Even as in the same fire gold glistens and straw smokes, so in the same fire the sinner burns and the elect is cleansed." Therefore the fire of Purgatory is the same as the fire of hell: and hence they are in the same place.
However, the actual text, which comes from Augustine's City of God, 1.8, does not specifically reference purgatory, but affliction in general:
For as the same fire causes gold to glow brightly, and chaff to smoke; [...] so the same violence of affliction proves, purges, clarifies the good, but damns, ruins, exterminates the wicked.
So it's questionable if Augustine meant to apply this idea to purgatory. For what it is worth, Aquinas decides that the most likely scenario is that:
the place of Purgatory is situated below and in proximity to hell, so that it is the same fire which torments the damned in hell and cleanses the just in Purgatory.