User blog comment:WitchKingofMorgul/Black Arrow theory/@comment-66.66.136.84-20130211145313

During the chapter "Inside Information" in The Hobbit, Bilbo converses with Smaug and tricks the dragon into revealing his belly. Smaug accidentally reveals a weak spot, which Bilbo notices and explains to the Dwarves. A thrush overheard this exchange and eventually told Bard about the weak spot during Smaug's attack on Lake-town. Bard successfully targeted the weak spot with the Black Arrow, killing Smaug.

This plot detail shows that the Black Arrow didn't NEED to be stronger than mithril because it didn't penetrate Smaug's armored belly (it only hit soft flesh). While your theory is an intriguing one, I don't think it's correct to assume that the Black Arrow was made of the same material as Gurthang, since there is no indication that the arrow is stronger than any iron/steel arrow.

I do think, however, that Tolkien intended for the two weapons to be thematically related, and with the publishing of his stories about Túrin, more astute readers (like yourself, I might add) can draw comparisons between the two.