If a woman simply leaves then she comes under the rulings of nashuz (rebellion) and the husband is not required to pay any financial maintenance to her.
Otherwise the law on spending for maintenance of a woman during her iddah because of divorce depends on the following factors:
If the woman is pregnant during her iddah, then she has a right to maintenance and lodging from the husband. This is regardless of whether the divorce is revocable or irrevocable or a khula'. That is because the verse below applies to all divorced pregnant women without any other conditions attached:
وإن كن أولات حمل فأنفقوا عليهن حتى يضعن حملهن
And if they should be pregnant, then spend on them until they give birth.
— Quran 65:6
Also because spending on the child is always obligatory for the father, and in this case spending on the fetus is inseparable from spending on the mother.
She has a right to maintenance and lodging from the husband. That is because she is essentially still locked in marriage and the divorce can be revoked at the choice of the husband without requiring remarriage.
If the woman is not pregnant and the divorce is irrevocable. There is a difference among the madhabs on this:
She has a right to lodging but not to maintenance. This is the stance of the Malikis and Shafi'is. It is based on the verse:
أسكنوهن من حيث سكنتم من وجدكم ولا تضاروهن لتضيقوا عليهن وإن كن أولات حمل فأنفقوا عليهن حتى يضعن حملهن
Lodge them [in a section] of where you dwell out of your means and do not harm them in order to oppress them. And if they should be pregnant, then spend on them until they give birth.
— Quran 65:6
As lodging has been granted to all divorced women while spending on maintenance has been conditioned to being pregnant.
She has a right to lodging and to maintenance. This is the stance of the Hanafis. Their evidence includes:
The tafsir of verse by Ibn Masud where maintenance too is applicable to all divorced women:
أسكنوهن من حيث سكنتم وأنفقوا عليهن من وجدكم
Lodge them of where you dwell (and spend on them) out of your means ...
Also because of the part of the verse which says:
لا تضاروهن لتضيقوا عليهن
do not harm them in order to oppress them
Whereas withholding maintenance is a form of harm.
And the tradition where Umar contradicted the hadith of Fatimah bint Qais and claimed that the Sunnah was to give the right to both lodging and maintenance:
لا نترك كتاب الله وسنة نبينا -صلى الله عليه وسلم- لقول امرأة لا ندرى لعلها حفظت أو نسيت لها السكنى والنفقة
We will not leave the Book of Allah and the Sunnah of our Prophet (ﷺ) for the words of a woman when we do not know whether she remembered or forgot. She is entitled to accommodation and maintenance.
Also because according to the Hanafis the reason for maintenance is the confinement of the woman due to the right of the husband. And a woman in iddah is still confined due to the right of the husband regardless of revocable or irrevocable divorce. Hence she has a right to maintenance.
She does not have a right to either lodging or maintenance. This is the stance of the Hanbalis. It is based on the hadith of Fatimah bint Qais:
فقالت طلقها زوجها البتة فقالت فخاصمته إلى رسول الله صلى الله عليه وسلم فى السكنى والنفقة قالت فلم يجعل لى سكنى ولا نفقة
She said: 'My husband divorced me irrevocably, and I referred my dispute with him about maintenance and accommodation to the Messenger of Allah (ﷺ). He did not grant me any accommodation nor maintenance.
According to the Hanafis, Shafi'is and Malikis the rule for a non-pregnant woman who received khula' is the same as their stance on irrevocable divorce.