But the question still is if relativistic calculations are needed in the GPS receivers or if the clock adjustment in the satellites is enough to compensate for relativity effects.
"Relativistic effects
The following section shall not provide a comprehensive explanation of the theory of relativity. In the normal life we are quite unaware of the omnipresence of the theory of relativity. However it has an influence on many processes, among them is the proper functioning of the GPS system. This influence will be explained shortly in the following.
As we already learned, the time is a relevant factor in GPS navigation and must be accurate to 20 - 30 nanoseconds to ensure the necessary accuracy. Therefore the fast movement of the satellites themselves (nearly 12000 km/h) must be considered.
Whoever already dealt with the theory of relativity knows that time runs slower during very fast movements. For satellites moving with a speed of 3874 m/s, clocks run slower when viewed from earth. This relativistic time dilation leads to an inaccuracy of time of approximately 7,2 microseconds per day (1 microsecond = 10-6 seconds).
The theory of relativity also says that time moves the slower the stronger the field of gravitation is. For an observer on the earth surface the clock on board of a satellite is running faster (as the satellite in 20000 km height is exposed to a much weaker field of gravitation than the observer). And this second effect is six times stronger than the time dilation explained above.
Altogether, the clocks of the satellites seem to run a little faster. The shift of time to the observer on earth would be about 38 milliseconds per day and would make up for an total error of approximately 10 km per day. In order that those error do not have to be corrected constantly, the clocks of the satellites were set to 10.229999995453 Mhz instead of 10.23 Mhz but they are operated as if they had 10.23 MHz. By this trick the relativistic effects are compensated once and for all." --
http://www.kowoma.de/en/gps/errors.htm