Eliot applauded Joyce for employing the 'mythical method' as a way of giving order to modern life. Examine and account for the uses of myth in the work of any one or two modern writers.
I think that one distinct way in which Joyce can be seen as embracing the "mythical method" is in A Portrait of the Artist as a Young Man. I think that the construction of Stephen Dedalus is an example of the Joyce's use of myth in his narrative. The Greek understanding of Daedalus is something that Joyce himself understood. Daedalus is one who creates labyrinths as an architect and is able to escape them. When imprisoned, he is able to fly above water and transcend the conditions that limit him.
Joyce appropriates this construction of myth in his development of his main character. Stephen Dedalus is one who seeks to transcend the conditions of the world around him. Issues of national identity, dogmatic spiritual identity, and familial ensnarements represent the symbolic towers from which he must escape. Joyce saw his character like "a winged form flying above the waves [...] a hawk-like man flying sunward above the sea, a prophecy of the end he had been born to serve." The development of Stephen as a character constantly in search of a "way out" and in search of his own labyrinth of identity in the form of his "non serviam credo" represents how Joyce appropriates the Greek understanding of Daedalus in his main character.
In the Greek myth, Daedalus is able to overcome the conditions that surround him, something that Joyce sees his main character able to accomplish. While the physical navigation of these elements is part of the Greek myth, Joyce internalizes this in the mindset of his Stephen. For Joyce, the "mythical method" contains a Modern approach. The physical journey is set against a more internal notion of voyage. What Daedalus must do in terms of physicality is something that Stephen must do internally. In being able to shift the voyage to within, Joyce shows a very skilled understanding of the Greek mythology in his A Portrait of the Artist as a Young Man.
Thanks so much for this answer. I am studying in London, England but come from an Irish family and am very interested in Joyce and Yeats, and Myths, Classical and Celtic. I have returned to study after teaching English for many years in China and have to say that I now find it difficult to get started on an essay, how to phrase the opening and closing. You have been most helpful.