"'Til Death Do Us Part"
Your divorce may be final, but your relationship isn't over.
When they're young, there are custody and visitation exchanges, birthdays, school plays, parent/teacher conferences, sporting events...and the list goes on and on. When they're adults, there are weddings followed by grandchildren (hopefully in that order), and the cycle repeats itself. Unless one of you walks away from your children completely, you'll have to deal with each other on some level for the rest of your lives. With that being said, doesn't it make sense to keep your divorce as civil as possible?
Most people can accept dealing with their ex, but I'm going to take it a step further knowing full well that what I'm about to say may cause you to stop reading. You will have a relationship with your ex's new partner. I didn't say it would necessarily be a good one, but it sure would make your life, and your children's lives, a lot easier if it is. No one wants to go to a birthday party and get into a fight that resembles a scene from "Real Housewives." On that note, sometimes you just need to make the best of a bad situation. That's life. You can't control what other people do, but you can control your actions. It takes two to tango, so sometimes simply refusing to engage is enough to diffuse a situation.
Most importantly, having a cordial or friendly relationship with your ex sets a positive example for your children and provides them with a more comfortable, stable environment.