Circuit simulates in Falstad but not in LTspice
LTspice needs ground to provide a reference voltage. You can put it where you like, but in some places the simulation may slow down due to the increased amount of 'above/below ground' activity. Generally it is best to put it where the real ground will be in your circuit.  

You should choose real components (eg. MURS120 diode, IRFZ44N MOSFET, Al electrolytic 2200 uF capacitor) rather than the default, to give them realistic parameters. Default parts can cause the simulation to become very slow or fail when they are used outside their normal parameters. 

Peculiar circuit configurations (such as trying to PWM a large capacitor) can also make the simulation slow. 


  When I simulate it in Falstad circuit simulator I get a result I
  expect, that is that voltage is regulated proportionally to the PWM of
  the generator.


You should not expect that. If the FET and diodes have low 'on' resistance then the capacitor will still charge up close to the peak rectified voltage, unless the PWM ratio is very low.