There have been 3 unions of elves and men that we know of; Beren and Luthien, Tuor and Idril, and Aragorn and Arwen. Beren and Luthien concieved a child (Dior) as did Tuor and Idril (Earandil). Elves do not need to give up their imortality to bear a child with a mortal. At the end of the First Age Elrond and Elros were given the choice of which race they wished to be accounted among. Elros chose to be counted among the mortals and Elrond among the Eldar (Elves). Elrond's children were also given this choice and Arwen chose mortaility. I'm not sure why she chose this, perhaps it was because in order to remain imortal she must remain with her father Elrond and she was unwilling to leave Aragorn.
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Yet the lies that Melkor, the mighty and accursed, Morgoth Bauglir, the Power of Terror and of Hate, sowed in the hearts of Elves and Men are a seed that does not die and cannot be destroyed; and ever and anon it sprouts anew, and will bear dark fruit even unto the latest days.
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