Arwen could've gone to the Grey Havens?

[Legolas]

But when King Elessar gave up his life Legolas followed at last the desire of his heart and sailed over Sea.

We have heard tell that Legolas took Gimli Glóin's son with him because of their great friendship, greater than any that has been between Elf and Dwarf. If this is true, then it is strange indeed: that a Dwarf should be willing to leave Middle-earth for any love, or that the Eldar should receive him, or that the Lords of the West should permit it. But it is said that Gimli went also out of desire to see again the beauty of Galadriel; and it may be that she, being mighty among the Eldar, obtained this grace for him. More cannot be said of this matter.

In this year [1541 in the Shire Reckoning] on March 1st came at last the Passing of King Elessar. It is said that the beds of Meriadoc and Peregrin were set beside the bed of the great king. Then Legolas built a grey ship in Ithilien, and sailed down Anduin and so over Sea; and with him, it is said, went Gimli the Dwarf. And when that ship passed an end was come in the Middle-earth of the Fellowship of the Ring.

Note the bold text: Legolas and Gimli did not set sail from the Grey Havens, nor did they accompany Gandalf there to say goodbye.