MD-11 Captain reported shortly after Takeoff Power set we received Master Caution Lights Flashing for a Level 1 Unable RNP alert. Per the briefing the takeoff was rejected. The flight crew returned to the gate for maintenance and the flight was cancelled.

Narrative

Shortly after Takeoff Power set we received Master Caution Lights Flashing for a Level 1 Unable RNP alert. Per our briefing the takeoff was rejected. We exited the runway and asked ATC for permission to stop on Taxiway 1 where we would investigate our issue. After setting parking brakes; I turned the airplane over to my First Officer instructing him to make sure we don't roll; and selected the FMS pages to the page for RNPs. The Copilot's was showing actual RNP well under the 1.0 required if I recall it was .01 which I assumed meant it was receiving a GPS signal. The Pilot's side however was showing over 3.6 over there times less the tolerance required of 1.0. I also observed the Navigation Displays (ND) and both sides were showing we were no longer fixing our position using GPS; but using IRS Nav only. I was fortunate to have Line Check Pilot (LCP) on the jump seat as our Relief Pilot this day and he concurred. I was considering deferring the GPS portion of the Nav system provided we had enough drift time available to us to operate in Class II airspace for the planned flight to ZZZZ. According to the flight plan and the drift times referenced in the FOM after also consulting the Operations this was doable. I also referenced the MEL and determined no (M) items would be required and then came back to the Maintenance section of the FOM to review the process of doing the deferral on our own with the assistance of Maintenance Control.I called Dispatch to inform them of our situation and to ask them to connect me with Maintenance Control. After speaking with Maintenance Control on speaker so my entire Crew could be involved with this process and be comfortable; I went into details on what I was viewing on the screens with them and my desire to go through the deferral process of the GPS to rely on the IRUs. At some point in the conversation Maintenance Control determined they wanted this looked at by Maintenance at the Gate; my entire Crew heard this and as a result we all decided the airplane needed to be returned to the gate. Once there; after showing the mechanic what we were observing; we got out of our seats and Station Manager Person A got in the left seat and the Mechanic got in the right seat. Person A stated to me this can be intermittent sometimes and will usually clear itself in the air and that it could just be satellite reception. To which I answered; then why is the First Officer's box working perfectly normally its using the same satellites. There was no answer. Person A had Person B; the Lead Mechanic for ZZZ on his cell phone some of the time. The Mechanic present decided to hard shut down the airplane. All power was removed. The Multi-Mode Receptor (MMR) circuit breakers were cycled there may have been others as well. They restarted the airplane all seemed good the mechanic went into the Centralized Fault Display System (CFDS) and said all the faults were clear and they thought they were done. The airplane was signed off as RTS (Return to Service). Then I observed the glideslope indicator on the NDs go back to IRS NAV once again. I pulled up the Position pages again and sure enough the FO's side was rock solid the Captain's side was drifting at a rapid rate. At this point ruling out satellite reception Person A told the Mechanic (at my suggestion) to swap out the AUX IRU into the #1 IRU slot. They did a full shut down again. The airplane appeared fixed. Person A seemed in his conversation with Person B to be suggesting the rejected takeoff was unnecessary. They are entitled to their opinion. It was wrong according to our training and the briefing we conducted. Shortly thereafter Person A left and the Crew began to get ready to leave again. We noticed the Captain's position drifting wildly once again to the extent it almost 4.0 in a matter of less than 15 minutes while the required RNP was 1.0. Additionally my First Officer pointed out to me that both NDs were showing L7.1 meaning the boxes thought we were 7.1 miles off our planned course. At some point I contacted the Dispatch again to let them know that although the airplane was signed off as fixed it most certainly was not. I wanted a Plan B. We asked for a number for the pilot on call and then we discussed it as a Crew. None of us (including a brand new LCP) wanted to bring a bird having Navigational issues to location if we could get it fixed here. The mechanic was pushing us to take it into the air cause when it got a signal he said it should update and be fine from there (there would be no signal updating over the Ocean). We pointed to the FOM that said it should be within 4.0 RNP for 6.2 hours after Initializing the IRU which we had done repeatedly during these multiple hard shutdowns. Eventually when the Mechanic realized we were not going anywhere he went down to the avionics bay again. When he came back up we asked him what he did this time that was different? He stated he 'reseated the Line Replaceable Units'. I asked what did that mean? He stated that it was not fully seated in its position and that could have affected the connections and its function. I am not a Mechanic nor Avionics guy. We didn't know if it would work so I came up with a plan to get it to ZZZZ regardless.The airplane had approximately 1.5 hours extra fuel over that required because it was originally planned to go very far off normal course around volcanic ash clouds. That NOTAM went away after we returned to the gate so Dispatch asked that we take our more direct route. I asked what the fuel burns would be at FL280 vs FL360-400 in case we had to come down before entering oceanic airspace due to this malfunction reoccurring. Turned out that a simple change to a closer alternate airport and we'd have enough fuel to make it even at FL280. Dispatch made the change; reissued a new flight plan with a new recall number and we got the airplane ready to go. We observed this time the glideslope combined GPS/IRS Nav seemed to be holding while we got the airplane ready once again. The fix held all the way to ZZZZ and the flight was uneventful. Of interest is this exact malfunction happened on this same airplane to me in ZZZZ on Date. I recall it because the prior crew documented the airplane exceeded the RNP10 requirements of oceanic airspace and this issue combined with another on Day 17 of my rotation eventually caused my First Officer to call Fatigue and I followed immediately thereafter due to a combination of events over multiple days piling up on each other.My entire flight crew became aware of this issue when the Master Caution lights began flashing at low speed during the takeoff roll.I believe this airplane has some sort of avionics issue that has been repeatedly addressed by shutting down the airplane; cycling circuit breaker; reseating computers (whatever that means); etc. over months without actually tracking down and fixing the issue in an attempt to keep the airplane flying and/or not order needed replacement parts. But that is only a guess. There may be another explanation. I'd refer the question to Maintenance.