Please help us help you and fill where relevant:
Your LimeSurvey version: 2.71.1+170927
Own server or LimeSurvey hosting: Own server
Survey theme/template: default
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We have noticed that when a user clicks Submit but has not answered some mandatory questions, LimeSurvey saves the data as an incomplete response. Is there any way to prevent this? It's important for us to comply with the General Data Protection Regulation (GDPR). We do not want any user data to be saved unless the user has checked the data consent box (it's a mandatory question).
Any advice on this?

Thank you so much!

You shouldn't use a more than four year old version if you comply with the GDPR.
Consider an update.
Here you have the option to display your private policy which the participant has to agree with before he is allowed to start the survy.

JOffm

Limesurvey saves responses to the database as soon as something has been answered. There is no way around it.

Now, the normal thing to do with a consent box is to have it as the first thing anyway. If you do so and they have not clicked the consent box, then in theory nothing that could be against the GDPR should be saved anyway.

And as Joffm noted correctly: If you really care about GDPR, then update your installation. Saving and storing data is NOT against the GDPR, as long as you do it the right way. And using a 4,5 year unpached installation is certainly not the right way.

While the replies regarding GDPR compliance may be valid, I think it would still make a lot of sense for survey admins to be able to opt out of responses being saved automatically.
Even if you don't care about the user not being aware of their data being stored (no matter what the privacy policy says...), it also makes a difference for people evaluating the survey. Yes, you can filter for completed responses, but why not introduce the ability to only record completed responses in the first place?

How do you think you will save the ones you want to keep? When do you consider a response incomplete? After how many minutes or hours? I think there is no general way to answer this. Limesurvey saves responses to the database, as soon as teh respondent clicks on "next". So these answers need to be stored in the database in any way, because Limesurvey does not know at that point if this survey will be completed or not.

So I think you will need to implement something yourself, that will delete incomplete answers after x minutes, hours or days. But this is something you will have to define. Or you just do it manually. But the way Limesurvey works is that all answers are stored in the database, as soon as a respondent moves to the next "page" of the questionnaire. I don't see any way how this could be changed easily.