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The electromagnetic counterparts to gravitational wave (GW) merger eventshold immense scientific value, but are difficult to detect due to the typicallylarge localisation errors associated with GW events. The Low-Frequency Array(LOFAR) is an attractive GW follow-up instrument owing to its high sensitivity,large instantaneous field of view, and ability to automatically trigger onevents to probe potential prompt emission within minutes. Here, we report on144-MHz LOFAR radio observations of three GW merger events containing at leastone neutron star that were detected during the third GW observing run.Specifically, we probe 9 and 16 per cent of the location probability densitymaps of S190426c and S200213t, respectively, and place limits at the locationof an interesting optical transient (PS19hgw/AT2019wxt) found within thelocalisation map of S191213g. While these GW events are not particularlysignificant, we use multi-epoch LOFAR data to devise a sensitive wide-field GWfollow-up strategy to be used in future GW observing runs. In particular, weimprove on our previously published strategy by implementing directiondependent calibration and mosaicing, resulting in nearly an order of magnitudeincrease in sensitivity and more uniform coverage. We achieve a uniform$5\sigma$ sensitivity of $870$ $\mu$Jy across a single instantaneous LOFARpointing's 21 deg$^{2}$ core, and a median sensitivity of 1.1 mJy whenincluding the full 89 deg$^{2}$ hexagonal beam pattern. We also place thedeepest transient surface density limits yet on of order month timescales forsurveys between 60--340 MHz (0.017 deg$^{-2}$ above $2.0$ mJy and 0.073deg$^{-2}$ above $1.5$ mJy).

LOFAR observations of gravitational wave merger events: O3 results and O4 strategy