The match between Aragorn, the future king of Gondor, and Arwen, the Evenstar of her people is a difficult one for many reasons. Their love faces many obstacles, including being tested by time and distance, as Aragorn has to leave to help the fellowship travel safely to Mount Doom for over a year, the love of another, in the form of the young maiden Eowyn of Rohan, and, arguably most significantly, the disapproval of Arwen’s father Elrond.
Elrond disapproves of Aragorn on many grounds, some more justifiable than others. In one way, it is because Aragorn is mortal, and will die, leaving Arwen bereft, in other ways, it is because he doesn’t believe that Aragorn is worthy of his daughter, the jewel of the elves, until he has proved himself by reclaiming the throne and reuniting the kingdoms. It is also because he knows that he will part to the Undying Lands at the end of the Third Age to be with Arwen’s mother there, and he doesn’t want to be half the world away from his daughter.
This is why he tries to convince Arwen to leave with him on the last ships sailing west to the Grey Havens. Despite the heartbreak that she would undergo in being parted from Aragorn, whom she loves dearly, Elrond genuinely believes that this would be the best thing for her. It is also important to note that at this time, no one knows what the after of the ring of power will be, and if the quest to destroy it will actually prevail.
This means that they are all facing the possibility of a future where all of the light is smothered out of the world, and the evil lord Sauron has total domination and corruption of all those who remain in Middle Earth. Therefore, when Elrond shows Arwen the vision of Aragorn’s death in order to convince her to leave, he really is just trying to keep her safe. However, this becomes tricky when Arwen realizes that he only showed her part of the vision.
As she herself accuses her father after he tells her “I looked into your future and saw death” that “there is also life! You saw there was a child. You saw my son!” This is difficult because it means that in convincing Arwen to journey to the Undying Lands, Elrond knew that he was preventing the existence of his own grandson. He did indeed see the child in the vision, and knew that he was destroying the possibility of this future happiness by making Arwen leave and sever her ties with Aragorn. As a father who genuinely loves his daughter, and wants the best for her, why would he commit such awful actions, when he was aware of the life that hung in the balance of his consequences? There are two strands to this answer.
The first is that, the future in which his grandson (named Eldarion) is born is just one possible future, a future in which the ring is destroyed, and Aragorn manages to claim the throne of Gondor. Although this is luckily the future that does actually happen, Elrond has no way of knowing this, and so he prefers to do something concrete now in sending Arwen away, rather than waiting to see the outcome of which of the many possible futures may come true in her path. The second reason is that it is far easier to try to protect the living, breathing daughter in front of him, whom he has loved and guided all her life, than to protect the possible, unborn grandchild that only exists in a dream. When it comes to the choice between the daughter he has now, and the grandchild he will never meet, because he must sail to Valinor, it is an easy choice to pick Arwen, and he would do so every time, even if he had more guarantee of Eldarion’s existence.
At the end of the day, Elrond truly only wants for Arwen to be happy. He feels that going to the havens, where she can be free of the suffering of the earthly world, will bring her true peace. This concept stems from Tolkien’s own religion of Catholicism, in which heaven is a sacred place away from struggle, and is also informed partly by Tolkien’s great friendship with C.S. Lewis, who also wrote about the world after death being the true light at the end of the tunnel in his Narnia books. But ultimately, Elrond does eventually give his blessing to Arwen and Aragorn, entrusting his most loved person and her future to Aragorn, because he realizes that it is the only way she will ever truly be whole.