Local Controller reported they cleared an aircraft to descend; but they went below the minimum altitude for the area and out of radio contact. Controller reestablished contact with the pilot and climbed them back to a safe altitude.

Narrative

Aircraft X came over to my frequency well outside my airspace at 17;000 feet requesting a ZZZ full stop. I gave him a pilot's discretion descent to 2;100 feet for the approach with a turn to the final approach course. By the time I noticed the rapid rate of descent and tried to stop the descent he was too low for radio coverage and may or may not have heard me. He continued to descend to an altitude that was below the MVA more than 40 miles from our radar site. After several attempts to re-establish communications he switched to my UHF frequency and I climbed him back up to 3;000 feet which is 300 feet above the MVA until he was within 40 miles of our radar site. At that point I brought him back down to 2;100 feet to clear an obstacle north of ZZZ1 airport. In the future I will avoid pilot's discretion descents that far outside of our airspace. If I do I will make sure that the altitude issued will meet the MVA outside 40 miles of our radar site in case of a rapid descent. Typically pilot's will slow their rate of descent with pilot's discretion so that they can stay at an altitude which will conserve fuel. That was not the case in this event.