Elenctics, in Christianity, is a division of practical theology concerned with persuading people of other faiths (or no faith) of the truth of the Gospel message, with an end to producing in them an awareness of, and sense of guilt for, their sins, a recognition of their need for God's forgiveness, repentance (i.e. the disposition to turn away from their sin)) and faith in Jesus Christ as Savior and Lord.
Johan Herman Bavinck (1964:221) explains that:
The term "elenctic" is derived from the Greek verb elengchein. In Homer the verb has the meaning of "to bring to shame." it is connected with the word elengchos that signifies shame. In later Attic Greek the significance of the term underwent a certain change so that the emphasis fell more upon the conviction of guilt, the demonstration of guilt. It is this latter significance that it has in the New Testament. Its meaning is entirely ethical and religious.
Though he uses an archaic terminology now generally eschewed by Christian missiologists (i.e. "heathendom"), Bavinck's explanation of Christian elenctics is still valid in the 21st century. To quote him at length:
When we speak of elenctics we do well to understand it in the sense that it has in John 16:8. The Holy Spirit will convince the world of sin. The Holy Spirit is actually the only conceivable subject of this verb, for the conviction of sin exceeds all human ability. Only the Holy Spirit can do this, even though he can and will use us as instruments in his hand. Taken in this sense, elenctics is the science which is concerned with the conviction of sin. In a special sense then it is the science which unmasks to heathendom all false religions as sin against God, and it calls heathendom to a knowledge of the only true God. To be able to do this well and truthfully it is necessary to have a responsible knowledge of false religions, but one must also be able to lay bare the deepest motifs which are therein expressed. This can actually occur only if one recognizes and unmasks these same undercurrents in himself. Elenctics is possible only on the basis of a veritable self-knowledge, which is kindled in our hearts by the Holy Spirit
Perhaps the most famous example of specifically elenctic literature in the history of Christianity is St. Thomas Aquinas' great work, Summa Contra Gentiles.