The way I see it is that Luthien had a choice to linger on or go when Beren died. It did not matter if she were mortal or immortal. It's the same thing with Arwen and Aragorn. When Aragorn died Arwen could have probably died with him, but she chose to linger on. Why, we may never know.

Elrond's gift of foresight is kind of an unfortunate thing and a tad bit confusing. He saw Arwen lingering on and told her so. When Arwen heard this couldn't she have just decided that she would not and so alter the vision her father had forseen? It's a whole deal between fate and free will. Elrond said her fate would be to linger on, but free will could have told Arwen to die with Aragorn.

It all has to do with the human mind...

"The Valar indeed may not withdraw the gift of death, which comes to Men from Iluvatar, but in the matter of the Half-elven Iluvatar gave to them the judement; and they judged that to the sons of Earendil should be given choice of their own destiny."

"...I should still be grieved because of the doom that is laid on us."
"What is that doom?" said Aragorn.
"That so long as I abide here, she shall live with the youth of the Eldar," answered Elrond, "and when I depart, she shall go with me, if she so chooses."
"I see," said Aragorn, "that I have turned my eyes to a treasure no less dear than the treasure of Thingol that Beren once desired. Such is my fate."
Then suddenly the foresight of his kindred came to him, and he said: "But lo! Master Elrond, the years of your abiding run short at last, and the choice must soon be laid on your children, to part either with you or with Middle-earth."
"Truely," said Elrond. "Soon, as we account it, though many years of Men must still pass. But there will be no choice before Arwen, my beloved, unless you, Aragorn, Arathorn's son, come between us and bring one of us, you or me, to a bitter parting beyond the end of the world."

We know from "The Tale of Arwen and Aragorn" that Arwen died in the spring after Aragorn died. (He died in the winter.)