I would like to do the analysis for the Crank-Nicolson method on a non-uniform grid for the parabolic equation with variable coefficients. I was able to prove everything for a uniform grid by energy methods for the results of stability, where I have used summation by parts to do estimates. Now that I turn to a non-uniform grid, I can't carry over the same proofs, since the summation by parts is different. Is there any reference on it? What arguments do people make to extend the proofs to that case?
I am 100% sure that there is literature from the 1970s and 1980s that show what you need. Take a look and see whether there's anything in Vidar Thomee's papers that helps.