At the nodes, there may be 'no displacement' but the force is high. (In a mechanical example- and there is always an equivalent for other waves). A standing wave is just the resultant of many waves travelling in different directions so energy will be travelling across the node for that reason.
IF you are looking for a 'deeper' reason how any energy can be passed from a zero displacement point (i.e. no Potential Energy) then I think you need to consider two cases.
1. The standing wave is starting up (the excitation has just been turned on) and the level of the standing wave is building up. In this case the nulls haven't formed yet as the left flowing energy is not the same as the right flowing energy.
2. The standing wave has established itself at its final level, at which point there is energy being lost at the same rate as energy is being supplied. I think that, as energy loss is involved, the phases of the reactive and 'resistive' parts, which are due to the loss mechanisms, will mean that you won't get perfect cancellation either so the nulls aren't perfect nulls, the depth of the null depending on the Q of the resonance. Energy is still flowing through the system.
sophiecentaur thanks for your reply.
Your description is ok for the " real world".
Suppose you have to introduce standing waves on the blackboard. You have two identical waves with equations y1 = Asin(kx-ωt) and y2 = Asin(kx+ωt) travelling at the same axis. There are two initial points where the waves arriving with time difference T/2. At these points we have nodes. In that case how we explain that energy passes?
Sorry for my poor english and thanks again.
Why do you think that the issue of energy flow is a problem? The same thing must apply at all times when waves cross each other - not necessarily originally from the same source or even at the same frequency. In a linear medium, the principle of superposition applies so there shouldn't be a problem. The E fields may nearly cancel at an E node, but, the H field resultant isn't zero.
What do you mean by "perfect"?
A rope is, by definition, a very real piece of stuff with a lot of friction. To maintain a standing wave, you need to be supplying energy constantly, to overcome the losses.
How would you measure that, in practice?
It may be an apparently very small displacement but, as you would be wiggling one end up and down, the arrangement wouldn't be all that stable.
But I still don't see your conceptual problem with the energy flow thing.
Well I think what OP is asking is that the kinetic energy of the piece of rope at the node is 0, since it has no velocity, or you could say that there is no acceleration, or just that it is not moving. Either way, how is it transferring movement, energy, acceleration?