[Voronwe]

Because she chose to marry Aragorn and remain in Middle Earth with him, Arwen became mortal. When she died in Lorien, she died a mortal woman, and thus the fate of her spirit after death was the same as Aragorn's.

After death, the spirits of mortals go to the halls of Mandos in Valinor, and after a short while they leave Arda (the world) forever. Where they go is largely unknown, though it is thought they go to meet Eru Iluvatar.

When elves are slain by either violence or grief, their spirits go to the Halls of Mandos in Valinor. After a period of waiting, some are reincarnated and allowed to live in Valinor, while others (such as Feanor) remain in Mandos until the end of the world.

As to the question of Arwen ever meeting up again with her Elven kin (such as Elrond) after her death, the answer is not known. While mortals leave the world when they die, Elves are bound to it until it ends. The fate of the Elves after the end of the world is uncertain. Tolkien writes a little on this subject in 'Athrabeth Finrod ah Andreth'. In the Athrabeth, Finrod Felagund says he believes Elves and Men will meet again after the end of the world, and that men will help to heal Morgoth's marring of Arda, though he admits it is no more than a hope.

-Voronwe