Having children is a right for both husband and wife. If the wife wants to have children but the husband refuses, it is permissible for her to ask for a divorce, because the need to have children is a real need, and there is nothing wrong with the husband divorcing her in the hope that Allah may enable her to have a child with a different husband. But we say that you should understand that having a child is a great blessing and that what you think is more enjoyable, for example, with no children, may be many times better with a child. Children are part of the adornment of this world, as Allah says (interpretation of the meaning)

“Wealth and children are the adornment of the life of this world. But the good righteous deeds that last, are better with your Lord for rewards and better in respect of hope”

[al-Kahf 18:46]. 

If your wife is righteous, religiously committed and of good character, then keeping her is better; perhaps Allah will bless you with a child who will be a delight to you both.  

There is nothing wrong with having children by artificial fertilization if that is done by taking the husband’s sperm and the wife’s eggs, fertilizing them outside the uterus then implanting the fertilized cells in the wife’s uterus, or taking the husband’s sperm and injecting it into the appropriate location in the wife’s cervix or uterus. 

There are other forms of artificial fertilization that are haraam. See the answer to question no. 3474. 

To sum up: It is permissible for you to divorce your wife so that she will be able to get married to another husband and have the children she wants. But if you keep her and help her to have children by means of permissible kinds of artificial fertilization, that is better. 

And Allah knows best.