After startup my battery is at 10.5 volts then after starting it up again it is 13.8 volts

You have a parasitic drain on your car, the battery voltage should not drop that low(unless it was sitting cold). You need to find the cause ,try using parking inside a garage(remeasure voltage next day) or use a clamp meter on amps and start pulling fuses from the fuse box to see the current go down , and isolate the problem (use electrical diagram from service manual for your car) .Note when you first unlock your car , the fuel pump/proximity circuits will start , you need to wait for that to settle down , then measure current whilst the car is off.

See https://www.youtube.com/watch?v=KF1gijj03_0 for what i mean.

RE: @Peter

Consider the proposition of a faulty wiring loom/device , if there is a short/open circuit (intermittent fault),pulling the voltage down , removing fuses may isolate the circuit if there is a short .All that we've so far established is, that the circuit operation normalizes after a restart. During that restart period connections could be tightened due to heat , or even have better connections due to the vibration jostling the connector/loom.I would starting checking connections to the big 3 and main battery connectors.

Playing off your point , the battery is charged using the regulator, via modulation by the ecu output , wiring/plugs could be damaged there as well , check voltage on alternator regulator plug/wiring loom (be careful of moving parts and belts with your hands/limbs/DMM leads).