A norm estimate for the ADI method for nonsymmetric problems

We give a norm estimate for the alternating direction implicit method for nonsymmetric elliptic convection-diffusion problems on a rectangular domain. We estimate a certain form of the iteration matrix in terms of the coefficients of convective terms and the mesh size. The norm is shown to be asymptotically of the form (1 − Ch) / (1 + Ch), where C is the same constant as in the symmetric case. We also show that the optimal size of the parameter is the same as in the symmetric case. As a consequence, we conclude that the convergence behavior is as good as that of the symmetric case and does not deteriorate as the size of convective terms grows. Numerical experiment shows that our analysis is sharp.