Can one repent from adultery, and is the Sharia-ordained punishment to be carried out on the adulterer?

Whoever falls into sin must offer sincere repentance, feel remorse, and resolve not to return to it. If Allah knows the sincerity of the repentance, He will forgive the sin and erase the transgression. Allah the Almighty says: "Say, 'O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.'" He must conceal himself and not inform anyone of his sin, while being warned that what he did is a grave sin and a great offense.