Good question! Tolkien wasn't happy with his first attempts at explaining this phrase linguistically, but this is what he finally settled on (notes paraphrased from Parma Eldalamberon #17, p. 16):
Mae, well g(I)'ovannen, well [art] thou met Tolkien proposed that the g in govannen is the contracted form g' of a pronoun gi 'thou' ... when this has undergone the necessary lenition following mae, while the participle is indeed govannen which itself has undergone lenition to 'ovannen in combination with the pronoun.
So basically it's Mae + Ci + Govannen, and the mutation goes as follows: Ci -> Gi Govannen -> 'ovannen
then elision kicks in with Gi + 'ovannen (two vowels together don't work in Sindarin so one of them has to go), making: G'ovannen.
I hope that makes sense, if not please ask any other questions!