The symmetric Sinc-Galerkin method yields ADI model problems

We show that when a symmetric Sinc-Galerkin method is used to solve a Poisson problem, the resulting Sylvester matrix equation is a discrete ADI model problem. We employ a new alternating direction scheme known as the alternating-direction Sinc-Galerkin (ADSG) method on illustrative partial differential equation boundary-value problems to document the exponential convergence rate that can be achieved. Unlike classical ADI schemes, direct numerical application of ADSG avoids the computation of iteration parameters, matrix eigenvalues and eigenvectors, as well as the use of the Kronecker product and sum.