The car may be "brand new", however, unless the battery has been replaced in the car at this point, the battery may be as much as 2-3 years old at this point.
When I see a 2012 model year car with 3k miles on it, when the model will be 2 years old 3 months from now, this tells me the car is not likely driven often and/or it sat a good portion of its life. If the battery is the original battery in the car, it is likely the battery ran flat and stayed flat for a long time somewhere along its journey.
How long have you owned the car?? 3k miles for me would be less than 3 months.
You have had it run fairly flat a few times since you have owned it if it has not started for you.
Running a battery flat will quickly shorten its lifespan and available capacity.
I would almost say I would want a new battery at this point anyway, it should be covered under warranty at this point? 2 year/24 month bumper to bumper warranty covers everything but tires, which are covered by separate coverage.
As with any battery/electrical problem, I would first question the battery. This should be very easy for a reasonable person to condemn, however, dealers are notorious for not having techs that fully understand electricity and electronics. This is because new cars usually have very few, if any electrical or electrical problems.
If you have the car back in your hands, then I would likely get the battery checked out at the local auto parts store if the dealer does not replace it. This should be free and another point of reference.
If the battery is draining overnight on a regular basis, then this should not be too hard for a reasonably competent tech to figure out, however, many dealers give cars this new to the lower level techs as cars with this few miles usually have really nic picky problems, squeak & rattles, things that are missing, fit and finish items. So the dealer may not be giving the car to a more seasoned tech.
At this point, if the dealer cannot find a problem, and it is happened a few times, then the dealer should be replacing a battery that is likely between 2-3 years old from the date of battery manufacturing,
Where are you located? Do you have other dealer options if the current dealer cannot sort this out quickly?
Good luck.
