A321 flight crew reported malfunction with FADEC. Diverted and landed uneventfully.

Narrative

Just after level off at cruise; received several ECAM messages relating to engine 1 FADEC (Full Authority Digital Electronic Control). Followed ECAM procedures and follow up measures. Contacted dispatch and they conferred with Maintenance Control. Decision made to divert to ZZZ. It would be an overweight landing; so we advised ATC. Reviewed overweight landing procedures. The engine continued to operate nominally the entire flight; descent; approach and landing. Cleared the runway; fire department checked the brakes and the all clear was given to taxi to the gate with trucks following.Cause: During initial pushback and engine start; we received the same 3 ECAM messages during the first engine start. Since we were off the gate; I contacted Maintenance Control by phone. Was told all three ECAM messages were no-go items and to return to the gate for maintenance action. Push-back crew towed us back to the gate. Maintenance worked on the aircraft and cleared all items and signed off the aircraft. Subsequent pushback; engine start; taxi; takeoff and climb were all normal. Shortly after level off the identical ECAM messages reappeared. Dispatch; working with Maintenance Control; had us divert to ZZZ.