Husband providing for wife during iddah

I was studying a marriage/divorce fiqh and one situation got into my mind and I couldn't find any answers to it in the internet, so I thought it will be heplful to ask it here and get an answer, so if anyone gets into this situation they will know an answer. As I know, if a couple get divorced, they need to wait for 3 periods (iddah) and after that the divorce will be finalized and during this period husband should keep providing for his wife. But what if the initiator was the wife, for example: it's not surprising in the modern world if the wife lefts his husband (moving to another country, doesn't want to live with him anymore, lefts to the parents and e.t.c.) and is demanding for a divorce. In such situation there is no option for a husband left other than give the divorce and there is no blame for the husband to do it, obviously, but one question everyone is forgetting in such situation is what about providing during this period? Because if he's not an initiator and they don't live together anymore, does he need to keep providing for her during this period? Because obviously providing while living together and not living together are two different things (economically and spiritually).