It is quite common to experience drowsiness during meditation. You become very calm and relaxed. That is exactly what is supposed to happen. Unfortunately, we ordinarily experience this lovely state only when we are falling asleep, and we associate it with that process. So naturally, you begin to drift off. When you find this happening, apply your mindfulness to the state of drowsiness itself. Drowsiness has certain definite characteristics. It does certain things to your thought process. Find out what. It has certain body feelings associated with it. Locate those.
This inquisitive awareness is the direct opposite of drowsiness, and will evaporate it. If it does not, then you should suspect a physical cause of your sleepiness. Search that out and handle it. If you have just eaten a large meal, that could be the cause. It is best to eat lightly before you meditate. Or wait an hour after a big meal. And don't overlook the obvious either. If you have been out loading bricks all day, you are naturally going to be tired. The same is true if you only got a few hours sleep the night before. Take care of your body's physical needs. Then meditate. Do not give in to sleepiness. Stay awake and mindful, for sleep and meditative concentration are two diametrically opposite experiences. You will not gain any new insight from sleep, but only from meditation. If you are very sleepy then take a deep breath and hold it as long as you can. Then breathe out slowly. Take another deep breath again, hold it as long as you can and breathe out slowly. Repeat this exercise until your body warms up and sleepiness fades away. Then return to your breath.

Novice Meditator