I listened to the LotR book and appendix, and understood that immortality was tied to the west, though also granted to the elves in Middle Earth (via blessing, was it?) until the end of the third age. If an elf wanted immortality after the third age, they would have to go to the west. Arwen decides to stay in Middle Earth to be with Aragorn and assumes there will be no ship to bear her back to the west. With no way to go back, she eventually dies.
After Aragorn's death, however, Legolas and Gimli go to the west, meaning there still were boats. With no reason to stay, why didn't Arwen go with them? Even if she would die in the west, why didn't she go back to be with her kinsmen instead of staying in Middle Earth, sad and alone?