God wants all people to live in peace and harmony with one another. Unfortunately when marital problems begin this peace and harmony is broken.
Christians should strive for peace in all of their relationships, not just in marriage. This means that everything that can be done, should be done to save a marriage.
Even when a partner has hurt their spouse greatly and there appear to be grounds for a divorce such as adultery (see “Is infidelity grounds for a divorce?” for more information) reconciliation should still be the goal.
No matter what harm has been caused by a spouse, a person can repent and seek forgiveness. God can save any marriage that both parties are interested in saving. God can also work to turn the heart of a spouse who has lost interest in their marriage and work to bring the couple back to a renewed love.
The Bible gives us one instance where a marriage should be ended for the sake of peace and harmony. 1 Corinthians 7:12-16 lays out an instance where a spouse should be allowed to leave.
12 To the rest I say this (I, not the Lord): If any brother has a wife who is not a believer and she is willing to live with him, he must not divorce her. 13 And if a woman has a husband who is not a believer and he is willing to live with her, she must not divorce him. 14 For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy.
15 But if the unbeliever leaves, let him do so. A believing man or woman is not bound in such circumstances; God has called us to live in peace. 16 How do you know, wife, whether you will save your husband? Or, how do you know, husband, whether you will save your wife?
In short, if one person is a Christian and the other is not, the Christian should not seek a divorce because they have some influence for good on their spouse and may yet lead them to the Lord. But if that spouse desires to leave the marriage, they should not feel obligated to remain in the marriage because they are a witness to their spouse. While this gives permission for a divorce to take place, it does not give permission to remarry. It is simply in the interest of peace that a divorce is allowed.
Unfortunately there are many instances of Christian couples who are not living in harmony. True Christians should do everything they can to save the marriage but often one person does not want to work things out. If a person is acting like an unbeliever, treat them like an unbeliever. If they refuse to seek the peace that God wants for the marriage, they are no better than an unbeliever.
Obviously this does not mean that the first time a spouse does not want to talk about marriage problems that they are acting as an unbeliever and should be allowed to walk away. Only in cases where a person obviously is not walking with the Lord despite claiming to be a Christian should this be considered and even then they should be the one to initiate a divorce.