Christ comments saying (ESV):
Matthew 5:31-32
31 “It was also said, ‘Whoever divorces his wife, let him give her a certificate of divorce.’ 32 But I say to you that everyone who divorces his wife, except on the ground of sexual immorality, makes her commit adultery, and whoever marries a divorced woman commits adultery.
Matthew 19:9
And I say to you: whoever divorces his wife, except for sexual immorality, and marries another, commits adultery.”
Mark 10:11-12
11 And he said to them, “Whoever divorces his wife and marries another commits adultery against her, 12 and if she divorces her husband and marries another, she commits adultery.”
Luke 16:18
“Everyone who divorces his wife and marries another commits adultery, and he who marries a woman divorced from her husband commits adultery.
I think that one could look at Christ's words and condemn remarriage after a divorce for anything other than adultery as being sin (adultery). However, some would put Christ's words in the context of one divorcing one's spouse for the express purpose of marrying another (a common sin of the Pharisees). Thus, Christ never mentions abuse or what may be other lawful grounds.
Paul expands on Christ's words saying:
1 Corinthians 7:10-1
10 To the married I give this charge (not I, but the Lord): the wife should not separate from her husband 11 (but if she does, she should remain unmarried or else be reconciled to her husband), and the husband should not divorce his wife.
I think the idea here is to try to reconcile. Therefore, a husband shouldn't immediately divorce his wife, should she live him. This goes for both spouses in most understandings. However, if reconciliation isn't possible and the husband has held out for an exceptional amount of time, a divorce might be warranted.
Paul also speaks of being abandoned by an unbelieving spouse:
I Corinthians 7:15
15 But if the unbelieving partner separates, let it be so. In such cases the brother or sister is not enslaved. God has called you to peace.
Here, Paul admonishes the believer that divorce is perfectly acceptable once an unbelieving spouse has abandoned the other. The brother or sister, according to Paul, isn't "enslaved"; or as the KJV puts it, isn't "bound" to the union. This is legal terminology used in Roman divorce decrees that permit remarriage. So many, feel that the believer is free to remarry in the Lord if abandoned by an unbelieving spouse.
Another angle has to do with remarriage being adultery. Adultery isn't an unforgiveable sin. Therefore if one (or both) in a second marriage brings the issues to God that led to the divorce and repents of them, the marriage stands.
Some acknowledge that the Bible doesn't speak about some topics (such as divorce and abuse) and therefore they call upon this text regarding Apostolic authority:
John 20:23
Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.
Thus, the spiritual authority of the elders can grant divorce on the grounds of abuse and speak forgiveness into the life of the abused, allowing them to remarry in the Lord.
Divorce and remarriage is a complicated issue that can be looked at from many different perspectives in Christianity. Clearly Christian brethren have, and do, disagree on the matter. I see it as a non-essential topic that we must wrestle with in accordance to our conscience, seeking justice and mercy for all parties involved in whatever interpretation we embrace.