What should a Muslim woman do who is married to a non-Muslim man who does not wish to convert to Islam, and she fears for her children, while she cannot easily separate from him, and what are the procedures for divorce in this case?

A Muslim woman marrying a non-Muslim is forbidden and invalid. She must separate from him and repent to Allah. A Sharia-compliant divorce is not required for the original marriage to be invalid, but she must confront the man about her mistake: either he embraces Islam and remarries her with a new contract, or he separates from her and officially divorces her to regularize her legal status. If he refuses, she should resort to the judiciary to obtain an official divorce. She should not marry anyone else until after one menstrual cycle to ensure she is not pregnant. It is advisable to seek help from an Islamic center.