Penalty can usually be omitted or converted in civil penalty if the victim decides to forgive or to accept compensation.
How are the regulations regarding adultery (according to any fiqh)?
No. The legal punishment of Zina can not be forgiven by the people. Once the case has been brought to the authorities and the required proof is established, then it is obligatory to carry out what Allah has commanded.
لا خلاف في حد الزنا والشرب والسكر والسرقة أنه لا يحتمل العفو والصلح والإبراء بعد ما ثبت بالحجة؛ لأنه حق الله تعالى خالصا، لا حق للعبد فيه فلا يملك إسقاطه
There is no difference regarding the hadd for zina, drinking and theft that they are not eligible for forgiveness, reconciliation or exoneration after proof has been established, for they are purely the rights of Allah and man has no right to drop it