Adverbs created by adding the suffix -mente to an adjective, keep the same graphic accentuation the original adjetive had.
So, as 'fácil' has a tilde, 'fácilmente' has it, too. But as 'raro' doesn't have, 'raramente' doesn't have either.
Your Basque friend is right and 'completamente' hasn't got any tilde.
This rule has been so since I remember, what means the last thirtysomething years.