Air carrier pilot reported experiencing expected GPS jamming in Georgia and Turkish airspace. Their GPS did not return as expected after reaching Budapest FIR airspace.

Narrative

We departed ZZZZ and it was a normal takeoff and climbout. Everything was normal until we reached the Tbilisi Fir boundary. As usual; our GPS was jammed. We continued on expecting it to come back around the Budapest Fir boundary. However; it did not. Eventually; I contacted Dispatch via SATCOM and told the of our problem with the GPS. I asked them to coordinate with Bird and Gander to make sure we could continue at our current altitude of 380. Dispatch wanted to confirm our ANP. We confirmed it was .51 at that time. Dispatch came back via ACARS and said we are good to go. Upon entering Bird airspace; we again verified that we did not have GPS; but that our ANP was now .81. At 20W; Bird descended us to FL320. 80 miles East of 30W we were told to cross 30W at 280. We complied. At that time; our FMS stopped updating times to the next waypoint and all fuel predictions. We were asked for and gave our 30 west again to Bird. After crossing 30W; we contacted Gander. It was time for my break. I came back up from my break after 50w and we were at FL400; econ fuel and zero on the cost index. It seemed we were not going to have enough fuel to safely fly to our destination. We chose to divert ZZZ for fuel. We landed safely in ZZZ.GPS Jamming is a regular occurrence through that part of the world. There just might be a procedure that another US carrier operating the Boeing aircraft is using to successfully solve the problem of the GPS not returning to normal operating modes upon entering the Budapest FIR boundary. If I was in charge; I just might look into it with Boeing and other carriers who operate through that area.