This question comes from Dr. D. Lane, a psychologist, who asks, "Based on scripture, should a Christian remain married to a non-Christian who is abusive, even though the abusing spouse wants to remain married? That's what I read in 1st Corinthians 7.12-13."
During the years of the writing of the Old Testament it is clear that Israelite women had few rights; indeed, women were little more than chattel, that is, property. A woman was not allowed to inherit property of any kind (thus a widow with no sons was truly desolate) and her role is portrayed as a child carrier and servant to the male.
According to the law, a man had only to pronounce publicly, "I divorce you, I divorce you, I divorce you" and the woman was no longer under the "protection" of her mate and was single, though ceremonially unclean until her ex-husband died.
In Jesus' day women still had few rights, though the influence of the Roman culture where women had many more rights was becoming vogue. Jesus challenged the conventional "wisdom" of the day and treated women with much more respect and with the dignity that should be accorded any human.
In speaking about the marriage/divorce question, when asked if a man could divorce his spouse for any cause and Jesus answered the question with words meant to protect the rights of the woman, "Except for the grounds of unfaithfulness" a man could not divorce his wife (according to some rabbis of the day burning food was grounds for divorce). Note that if the man suspected unfaithfulness by his spouse, according to the Israelite law, he could demand an abortion if she were pregnant (Numbers 5), and if he had proof of her unfaithfulness he could legally have her put to death, though Roman law prohibited this. In any event, Jesus' answer was meant to offer some measure of protection for the woman.
Paul dealt with a different marital matter in his letters--should you divorce a non-Christian if you are a Christian? His question seems to be couched in the frame of Ezra's command for all Israelites to divorce themselves from foreign wives (Ezra 10).
In defense of Israel, it should be understood that not all agreed with this decision. The book of Ruth seems likely to have been written in response to Ezra's command. In Ruth it is shown that King David's lineage can be traced back to a "foreigner," that is, both Ruth who was a Moabite and Boaz who was the son of a Canaanite prostitute (Rahab--Matthew 1.5). In this setting, Paul again makes an attempt to protect the woman's rights, as well as protect the sacredness of marriage by insisting marriages remain intact.
But what if the relationship is abusive? Unfortunately, scripture doesn't address every specific case, instead the timeless stories and principles must be viewed and reviewed. Though Jesus "prohibited" divorce except for unfaithfulness, I believe a case can be made that unfaithfulness to the spirit of marriage could be just cause for divorce.
Indeed, Paul insists that a husband should treat his wife in a Christ-like manner and in a manner like he would want to be treated (Ephesians 5.25-29) and the wife should respect her husband (Ephesians 5.33). When one spouse abuses the other, certainly there has been a breach of the marriage vows and the spirit of marriage that both Jesus and Paul spoke of.
Beyond this, as already shown, Jesus and Paul were concerned with protecting the rights of the woman. As such, they clearly would have supported a separation or divorce, especially if the abuser refused to get help and his (or her) part in keeping the sanctity in the marriage.