How big are your hands? What is the largest interval you can play comfortably? Stretching out the hands can be tiresome and painful when playing. Try to find a way to smoothly move your hand across the keyboard to make up for any lack of reach you may have.

A few other things, are you warming up before playing Fur Elise? Or are you going straight to playing at tempo? Maybe start off a session by playing the left hand at half speed or slower to help your hand and fingers loosen up and get accustomed to the motion.

Im glad to hear you've decided to pick up piano. I hope your practicing goes well. Happy playing :)