“Lady Undómiel,” said Aragorn, “the hour is indeed hard, yet it was made even in that day when we met under the white birches in the garden of Elrond where none now walk. And on the hill of Cerin Amroth when we forsook both the Shadow and the Twilight this doom we accepted. Take counsel with yourself, beloved, and ask whether you would indeed have me wait until I wither and rail from my high seat unmanned and witless. Nay, lady, I am the last of the Númenoreans and the latest King of the Elder Days; and to me has been given not only a span thrice that of Men of Middle-earth, but also the grace to go at my will, and give back the gift. Now, therefore, I will sleep.
“I speak no comfort to you, for there is no comfort for such pain within the circles of the world. The uttermost choice is before you: to repent and go to the Havens and bear away into the West the memory of our days together that shall there be evergreen but never more than memory; or else to abide the Doom of Men…”
“Nay, dear lord,” she said, “that choice is long over. There is now no ship that would bear the hence, and I must indeed abide the Doom of Men, whether I will or I nill: the loss and the silence. But I say to you, King of the Númenoreans, not till now have I understood the tale of your people and their fall. As wicked fools I scorned them, but I pity them at last. For if this is indeed, as the Eldar say, the gift of the One to Men, it is bitter to receive.”
“So it seems,” he said. “But let us not be overthrown at the final test, who of old renounced the Shadow and the Ring. In sorrow we must go, but not in despair. Behold! we are not bound for ever to the circles of the world, and beyond them is more than memory, Farewell!”
“Estel, Estel!” she cried, and with that even as he took her hand and kissed it, he fell into sleep. Then a great beauty was revealed in him, so that all who after came there looked on him in wonder; for they saw that the grace of his youth, and the valour of his manhood, and the wisdom and majesty of his age were blended together. And long there he lay, an image of the splendour of the Kings of Men in glory undimmed before the breaking of the world…”
“But Arwen went forth from the House, and the light of her eyes was quenched, and it seemed to her people that she had become cold and grey as nightfall in winter that comes without a star. Then she said farewell to Eldarion, and to her daughters, and to all whom she had loved; and she went out from the city of Minas Tirith and passed away to the land of Lórien, and dwelt there alone under the fading trees until winter came. Galadriel had passed away and Celeborn also was gone, and the land was silent.
There at last when the mallorn-leaves were falling, but spring had not yet come, she laid herself to rest upon Cerin Amroth; and there is her green grave, until the world is changed, and all the days of her life are utterly forgotten by men that come after, and elanor and niphredil bloom no more east of the Sea.”
– J.R.R. Tolkien, The Return of the King, LoTR Appendix A, The Tale of Aragorn and Arwen