What is the airflow across and below an elevator?
When an elevator moves (trailing edge) up, it partly moves in the way of the airflow. That means pressure goes up.The elevator is an airfoil itself, in the shade of another airfoil. If it moves up, the same thing happens when a wing moves trailing edge up: lift works downwards.

Image source

Picture above shows an airfoil with trailing edge down: lift is upwards. 


Upper aft side of the airfoil: air only moves faster if the surface moves out of the way of the stream, if the shape moves away gently enough the air still tries to follow. Note that the nose of the shape had pushed air out of the way, so there is less air along the upper aft side of the picture: lower pressure. 
Lower side of the airfoil: air tries to stream like it did before, but now there is a shape in the way of the airstream. The air needs to stream through a narrower stream tube: higher pressure. When the stream bumps into the airfoil at a shallow angle, it gets deflected downwards: impulse lift.