Using a vortex element method with an O(N) algorithm, how can I find out the speed and gradient of speed from the induced potential without using a grid ?

You'd have to obtain the gradient of the integral equation for the potential to get the velocity, and then another gradient to evaluate the velocity gradients. Then you'd have to apply the same multipole expansion techniques that are applied to the potential to the new equations.

Check out the literature on fast multipole expansion methods. You can get started by surfing through the net.

Adrin Gharakhani