Does God's knowledge increase in time? Does God today have different knowledge than yesterday?
In other words, did God not know yesterday what will happen today?

I'm mostly interested in Catholic and Protestant traditions
If God is increasing in knowledge, then God is by definition changing.  As such, technically you would be rejecting the "immutability" of God, the doctrine that says God never changes. 

While proponents of Open Theism would be more than happy to reject it, these are a fairly minor subset of Christians. Indeed, for the Open Theist, God is able to form relationships with humans only if he doesn't know what is happening in that relationship. And for the Open Theist, God's whole creation is a journey of discovery of the consequences of his Creation, hence God's knowledge is increasing.  

That said, most Evangelicals and technically all Catholics would reject all of this.  In a nutshell, if God is mutable, he cannot be omniscient nor can he be omnipotent. Many heresies spring from Open Theism and rejection of God's immutability.  Amongst other mainline Protestants, you'll most likely find more reject the assertions of Open Theism than those who would accept it. Historically, the immutability of God was the point - that God never changes - and to deny it would be heresy.

There is much more in this question:

What is the basis of the argument that Open Theism is heretical?

and this one:

Can God Change his mind?

If you'd like to understand the relative merits of the positions.