you have to equip Paine as a dark knight (9999hp mastered charon) and Yuna as a gunner with at least 7000hp, knowing trigger happy lv. 3, rikku as a white mage with at least 1500hp. Every character should be around lv. 55. I beat black elemental on the second try. For beating black elemental a chest where black elemental was containing catnip.

Build up a good defense with shell and protect, and have paine just use charon. If you cant beat it with charon have yuna use trigger happy or tableturner to beat it.