Chebyshev points and finite difference method

Hello.

I'm not a mathematician so it should not be strange that I have a problem.

I'm doing some research on interpolating polynoms.

Using the finite element method I obtained values in nodes, lets say displacements, and first used Lagrange polynomial interpolation with wich I got some polynom that behaves bad in the ends of interval.

But when I use Chebyshev points I get better results.

And here comes the problem. Finite difference has the same distance between points but Chebyshev points do not have, so the conclusion should be that it cannot be compared.

Or could they?

My question is that, could I compare this two methods? Is there any article that says how to treat finite difference with Chebyshev points.

Thanks

Best regards