Audi A4 - losing coolant rapidly but no leak?

Thanks for the clear explanation. I should say that I am pretty much clueless about cars (ask me about something simpler like the human body or my bicycles!), so it’s quite possible some of these have been done by my mechanic already. I don’t have any reason at the moment to doubt his expertise but will keep it in the back of my mind (given that this is still ongoing).

Nonetheless, I talked briefly with him this morning and will drop off the car tomorrow with all of your suggestions and see what he has to say. I’ll circle back in a few days and let you all know what happened.