Imam Khomeini and other marja’s have said: “If a woman becomes pregnant through adultery, in the case of her or the man or both being Muslim, she cannot abort the baby.” In this case the baby is the woman’s child, the only thing is that the baby won't inherit from the mother.
This question was also sent to the offices of some of the maraje’ and the following are their responses:
The office of the grand Ayatullah Khamenei:
Abortion is haram and the circumstances stated in the question aren't an excuse for its permissibility.
The office of the grand Ayatullah Makarem Shirazi:
It isn't permissible unless there is a very big and important problem that can only be solved through such.
The office of the grand Ayatullah Sistani:
If there is fear of the mother being harmed or if it entails severe hardship and difficulty it is permissible, but it must be done before spirit has been breathed into the fetus.
The office of the grand Ayatullah Safi Golpaygani:
The excuses stated in the question in no way make abortion of the baby permissible.
The office of Ayatullah Mahdi Hadavi Tehrani:
If keeping the baby has a considerable harm for the mother or entails severe difficulty and hardship for her, abortion will be permissible before spirit has been breathed into it, and as long as one isn't sure of the harm or difficulty, and there aren't considerable chances of such happening either, it will remain haram.
In order to have been cautious, it is wajib to refer to the qualified judicial authorities and to get verification from expert practitioners before doing so.
For further information, see:
The ruling on the adulterer and adulteress and the child whom is born from adultery, Question 2030 (website: 2397).