What i always do to make sure the charger unit is working is to measure across the battery terminals first without being connected to shore power. The battery terminal readings will be 12.6-7VDC if battery is fully charged.
Then I connect up to shore power and make the reading again across the battery terminals and it should read 13.6VDC or higher if you have a smart mode converter/charger hooked up.
This tells me I have a working working converter/charger unit and the DC cables patch is connected from the battery terminals to the DC OUTPUT of the converter/charger... i.e. all connections and the fuses are working normal between the Battery terminals and the DC OUTPUT of the converter/charger.
In your case it appears the battery terminal of 11.2VDC is almost a dead battery... a 50% state of charge battery reading should be around 12.0VDC...
If all you have charging is 13.6VDC then Progressive Dynamics says it will take 40-hours to return the battery to 90% of full charge and 78-hours to reach full charge. One downside to charging with 13.6VDC for this long of time may result in some boiling out of battery fluids... This is one main reason why going with SMART MODE charging modes is a hugh plus for maintaining good battery maintenance...
Roy Ken