Even if Elrond knew that the hobbits were coming despite not being told so, why would Arwen be sent. She was the daughter of Elrond, (the lady of Rivendell) so why would her safety be risked especially when the Nazgul were near when there would have been plenty of others better suited to finding them and going against the Nazgul, such as warriors like Glorfindel.
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His sword was long his lance was keen
His shining helm afar was seen
The countless stars of heavens field
Were mirrored in his silver shield
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