A319 Captain reported the aircraft elevator made momentary uncommanded pitch up /down movements. This event occurred about 15 times between the destination airport and the diversion airport. The flight crew landed at diversion airport with no issue.

Narrative

During this flight; I was the Captain/Pilot Monitoring while the First Officer (FO) was Pilot Flying. During flight the aircraft elevator made momentary uncommanded pitch up /down movements. This was felt in both side sticks and was also observed by momentary elevator pitch up/down oscillations on the F/CTL page. The VSI would show +/- 500 FPM momentarily during these events; but aircraft would gain/lose less than 100 feet. This event occurred about 15 times between ZZZ and our diversion airport in ZZZ1. During pre-flight logbook inspection; we noted that on the last leg of the previous day; the aircraft had a logbook entry that stated 'F/CTL ELEV SERVO DISPLAYED IN FLIGHT LEFT ELEV GREEN HYD SYSTEM.' The entry was signed off with 'Reset of ELAC...' Due to the previous day history; and our concern that the condition appeared to be getting worse and more frequent; the flight crew determined that the safest course of action was to get the aircraft on the ground as soon as possible to avoid a potentially serious uncommanded flight control event. No emergency declared. I instructed the FO to continue with Pilot Flying and ATC communication duties while I briefed the Flight Attendants; passengers and Dispatch. With my concurrence; the FO requested a diversion to our nearest airport at the time; which was ZZZ1. ATC gave us immediate assistance and vectoring; therefore there was no delay to our diversion. Using the QRH; I briefed the Flight Atttendants that we were making a precautionary landing in ZZZ1. I then made a PA to the passengers regarding the same. I contacted Dispatch as outlined in the FOM to advise of the diversion. The FO and I then reviewed the Overweight Landing procedure; as were close to being overweight on touchdown. The FO made a good suggestion as to starting the APU inflight to help with some additional fuel burn. We also slowed and extended the gear early to burn more fuel. We landed [Runway] XXR at ZZZ1 with no issue and about 600 pounds below Max Landing Weight. The aircraft also has an ACARS in STBY MEL. This rendered ACARS messaging with Dispatch inoperative. Looking up dispatch frequencies took some extra time; and was an extra distraction. This did not however cause any delay in our ability to divert.I believe that this was a mechanical issue with the aircraft; that we all know does occur from time to time. I don't know maintenance procedures; and I am confident that our maintenance team handled the previous write-up of 'F/CTL ELEV SERVO' according to procedure; but perhaps if there was more of an inspection than a 'reset' maybe another underlying issue would have been spotted.The aircraft also has an ACARS in STBY MEL. This rendered ACARS messaging with dispatch inoperative. Looking up dispatch frequencies took some extra time; and was an extra distraction. This did not however cause any delay in our ability to divert.I believe that this was a mechanical issue with the aircraft; that we all know does occur from time to time. I don't know maintenance procedures; and I am confident that our maintenance team handled the previous write-up of 'F/CTL ELEV SERVO' according to procedure; but perhaps if there was more of an inspection than a 'reset' maybe another underlying issue would have been spotted.