Voltage control with PWM-signal

Hello,
I now have the circuit set up like in Tom's diagram pwmreg.jpg (Thank you, Tom!), which was very similar to what I had previously... In addition to what is shown in the jpg file, I have two LEDs (parallel to one another) in series with the 1000uF Capacitor, with their negative poles connected to the ground. I also have a lead going from the negative pole of the 1000uF capacitor into my container of water. Another pole is also in the water, connected to the positive output of my variable power supply (V_out +). I am measuring the voltage (with a DMM) between the negative pole by the water jar (essentially right after the capacitor) and V_out+ of the variable power supply. This measured voltage is less than that from the power supply, but changes proportionately to the voltage set at the power supply and does not seem to be being controlled by the arduino/Mosfet. If I pull the positive pole (anode) out of the water jar the LEDs dim and the measured voltage drops. After doing this several times, the LEDs both broke. Putting a new LED back in makes the measured voltage rise again. Without an LED as a load, the voltage measured voltage is closer to what I actually want, but still changes proportionately to the voltage set by the power source.

I'm thinking that the problem might have something to do with the ground. Everything that is grounded is connected to the GND on the Arduino, but nothing is connected to the ground of the variable power supply. Should the other GND pin of the Arduino be connected to the ground of the variable power supply in order to give them a common reference?

I have the arduino plugged into an outlket with a 12V transformer. When I look at this plug, it doesn't have a ground pin. What are the ground pins from the arduino actually connected to?

all the best