A TRACON Controller reported an aircraft descended below their assigned altitude and flew below the Minimum Vectoring Altitude.

Narrative

Aircraft X was inbound to ZZZ IFR coming from ZZZ1 airspace. ZZZ1 called to apreq the aircraft direct to ZZZZZ; which is on the RNAV [Runway] XX approach. I approved them to go direct to the fix but told the ZZZ1 controller to advise the pilot that there were no instrument approaches into ZZZ. The pilot checked on with me asking why there were no instrument approaches into ZZZ. I informed him that it is in the NOTAMs and that the NOTAMs have had it for over 6 months. I advised him that he could track the RNAV approach inbound; but that he would have to cancel IFR or go for a visual approach. I gave him the lowest altitude possible for the MVA. I started to call the airport for him; and he said he needed lower. I informed him that I could not give him lower for the MVA and that he would have to cancel or climb up to 030 and try again. He did not cancel IFR and continued to descend on his own. I issued a low altitude alert and told him to check his altitude because he was at 017 and assigned 024. He continued and then eventually cancelled his IFR. I waited and had the CIC (Controller in Charge) brasher the pilot after he landed at the airport. His altitude and proximity to the airport were such that I did not feel it would be safe to brasher the pilot on frequency. I spoke with the OM after the incident and he told me about the conversation he had with the pilot. The pilot called a little later to pick up his IFR clearance for his return trip. At this point; the sector was being split and I told him to change to a different frequency and monitor. I then briefed the controller opening up that he was on frequency looking for his clearance. I intended to file the report regardless; but I was informed a few hours later that he was somewhere outside of Location X and was okay. Pilot education is really important for this instance. The pilot behaved as though he was on a VFR flight plan and did not listen to my instructions. He also clearly did not thoroughly go through the NOTAMS and weather for the area.