Center Controller reported an aircraft deviated from their assigned altitude and route to a non towered airport and descended below the MVA.

Narrative

Aircraft X was initially attempting to land at ZZZ. The clouds were too low and there are no approaches to this airport so the Pilot changed the destination to ZZZ1. I informed the Pilot that the bases of the clouds had been around 2500 ft. all day and that the Pilot would most likely need to shoot an approach. The Pilot requested the RNAV XX and I issued direct to the IAF. The Pilot never turned on to the approach and asked the Pilot what he was doing. The Pilot said he wanted the Visual Approach again. I cleared the Pilot direct to ZZZ1 at that point and called the airport to the Pilot. A minute later the Pilot descended below the MVA. I issued a low altitude alert. The Pilot continued to descend and I informed the Pilot he needed to climb. The Pilot climbed back to 3000 ft. and then almost immediately descended below the MVA again. At this point the Pilot said he had the airport in sight so I cleared the Pilot for the visual approach. I attempted to issue the brasher statement but the Pilot was trying to cancel and get off frequency. We could not reach anyone at ZZZ1 to try to get the Pilot to call Center.