v. 11 intends to explain v. 10 by pointing out that, just as the Spirit searches out the deep things of God and knows them, the Spirit is in fact the only one privy to such knowledge and it is only by the impartation of the Holy Spirit that we are vicariously capable of being cognitively enlightened concerning the things of God. Paul goes on to confirm what he has just said in v. 10 by drawing an analogy between the human spirit and mind and God's Spirit and his own mind. The analogy ought not to be pressed too literally, since this may result in some odd theology. It certainly ought not to be used as a prooftext for a theologically significant trichotomous view of the human person, in analogy to the Trinity, since this is quite far from what Paul intends. Just as no one knows one's own mind except oneself, so also does no one know the thoughts of God excep tthe Spirit of God. Albert Barnes' comments on 1 Corinthians 2:11 is worth quoting at length here:
Save the spirit of man ... - Except his own mind; that is, himself. No other man can fully know them. By the spirit of man here, Paul designs to denote the human soul - or the intellect of man. It is not to be supposed that he here intends to convey the idea that there is a perfect resemblance between the relation which the soul of man bears to the man, and the relation which the Holy Spirit bears to God. The illustration is to be taken in regard to the point immediately before him - which is, that no one could know and communicate the deep thoughts and plans of God except his Spirit - just as no one could penetrate into the intentions of a man, and fully know them, but himself. The passage proves, therefore, that there is a knowledge which the Spirit has of God, which no man, no angel can obtain, just as every man's spirit has a knowledge of his own plans which no other man can obtain; that the Spirit of God can communicate his plans and deep designs, just as a man can communicate his own intentions; and consequently, that while there is a distinction of some kind between the Spirit of God and God, as there is a distinction which makes it proper to say that a man has an intelligent soul, yet there is such a profound and intimate knowledge of God by the Spirit, that he must be equal with him; and such an intimate union, that he can be called "the Spirit of God," and be one with God, as the human soul can be called "the spirit of the man," and be one with him.
In all respects we are not to suppose that there is a similarity. In these points there is - It may be added that the union, the oneness of the Spirit of God with God, is no more absurd or inexplicable than the union of the spirit of man with the man; or the oneness of the complex person made up of body and soul, which we call man. When people have explained all the difficulties about themselves - in regard to their own bodies and spirits, it will be time to advance objections against the doctrines here stated in regard to God.
Let us compare this passage and Barnes' comments on it with a brief exposition of Romans 8:26-27 in order to more fully illustrate this point.
26 Likewise the Spirit helps us in our weakness. For we do not know what to pray for as we ought, but the Spirit himself intercedes for us with groanings too deep for words.
The word translated "makes intercession", ὑπερεντυνγχάνει is legal language used to speak of an advocate aiding someone in a court. In other words, the Holy Spirit pleads on our behalf to God just as Paul says Christ does in the very same chapter(Rom. 8:34). That we can trust in the Spirit's pleading and praying for us to God is clear from the very next verse:
27 And he who searches hearts knows what is the mind of the Spirit, because[g] the Spirit intercedes for the saints according to the will of God.
God knows what the Spirit is thinking and what he wants because he knows that the Spirit knows what God the Father is thinking and what God the Father wants. But this "knowledge" is not bare awareness, but refers to God's approval of the Spirit's intercession for the saints. The reason he "knows"(read: approves of, and regards effectually) the intercession of the Spirit for the saints is because the Spirit intercedes perfectly in accordance with God's will. The Spirit knows the desires and counsels of God the Father(1 Cor. 2:10, 11) and in accordance with this perfect knowledge, renders perfect obedience to God in his perfect intercession on behalf of the saints. Such prayer God must "know", that is, approve(cf. Gal. 4:9, 1 Cor. 8:2-3) since it is according to his will, just as he must regard Christ's requests as he makes intercession for his elect(Jhn. 11:22, 42, chapter 17, Rom. 8:34).