King Elessar (as Aragorn was known when he took up the throne) and Arwen did indeed have children. The only named child is Eldarion, son of the couple. Eldarion became King of the Reunited Kingdom or Arnor and Gondor after his fathers death in the year 120 of the Fourth Age. Eldarion would have inherited his kingship within the traditions of Numenor his father gave him the tokens of his rule, and then gave up his life willingly.