How do I analyze the error for the Crank-Nicolson method on a parabolic PDE?

Question

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?

Answer 1

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.