Since you know it is invasive then you should not be surprised to learn that yes, it will certainly come back, no problem. For the best long term look, I would run over it with a lawnmower or use a weed whacker.

get a weed eater. use it as an edger to clean up the edges. run a mower over the long flats. it'll come back. in fact, you'll probably never get rid of it.

whatever you do, under all circumstances, DO NOT LET IT CLIMB YOUR TREES, if it hasn't already killed and downed all of the nearby trees. when ivy grows up trees, it suffocates the tree and will kill it (holds moisture to the bark, etc). this is a huge problem in the Buckhead area in particular from what i've seen (people too rich to care, i personally believe, whether it's legitimate or not)

if it's already climbing the trees, cut the vine at the ground, then ~3ft up the tree and remove that entire section of the vine. let the top half die; you can pull the remainder of the vine off the tree it off easily once 3-4 months later.

if everyone understood the dangers of ivy and were responsible to keep their ivy contained and off of nearby trees, i honestly believe it WOULD NOT have the negative reputation it's acquired. i truly see it as an invasive treekiller, though, because that's what people've let it become...

Yesterday, ny wife began her battle to rid the yard of english ivy. We both figure it will be a long tough fight. The war has begun!

Good luck to you! I am 99.9% rid of ours! It IS possible, just be persistent.

Mowing and weed whacking doesn't do much, in my opinion. I just grab the runners and pull and pull and pull and pull... Like satellitehead said: don't let it climb your trees!

Also, I'd suggest wearing gloves when playing in the ivy. After two years of battling the stuff barehanded, I've developed a contact allergy to the stuff and break out in a rash a million times worse than poison ivy.

Just to clarify, Re: weed whacking and mowing, that was just in response to the original poster's question about trimming.