After Elsa's powers are revealed to the people of Arendelle Elsa runs away and climbs up to the shoulder of an impressive mountain crag. Then she uses her magic to build a majestic ice palace. It is unclear how long all this takes, but I think it is safe to say that Anna doesn't reach the palace until early afternoon.
When Anna does confront Elsa in Elsa's new ice palace Elsa seems to think she will be able to live alone in the ice palace for the rest of her life. However she would clearly need some support in the form of food at a minimum. By this time the adrenaline of the flight from Arendelle should have worn off, and Elsa surely would be getting hungry. We know both girls love the smell of chocolate, but that doesn't signal in my mind that Elsa must eat.