Convergence analysis of the ADI scheme for parabolic problems using discrete harmonic functions

For the heat equation on a rectangle, we consider the finite difference ADI method without a perturbation term on vertical sides for the intermediate solution. Using stability results of Andreev [1, 2] for the discrete harmonic function we prove, except for a $\sqrt{\ln(1/h)}$ factor, the second order bound that is stated without a proof by Samarski [8].