The V groove is the correct way to go but then you must put fiber tape over the crack and then use durobond instead of common joint compound. Feather it way out and you should be fine. Most times a crack in that location means they put a joint at that spot instead of wrapping the corner.

What you describe is a very common problem caused by a joint occurring at the corner. The cold temps cause stress to occur, cracking the joint. The sure solution is to remove drywall around your door and replace so there is no joint at the corner, but that is a fairly big job. A quicker solution that might work would be to retape using high-strength crack repair tape (See Link Below). I repaired a similar crack a year ago with this tape and it is still holding up.

Here is a link that might be useful: Repair Drywall Cracks