Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems

In this paper, we investigate a Galerkin-spectral method, which employs coordinate stretching and a class of trial functions suitable for solving singularly perturbed boundary value problems. An error analysis for the proposed spectral method is presented. Two transformation functions are considered in detail. In solving singularly perturbed problems with conventional spectral methods, spectral accuracy can only be obtained when N = O(ε^-γ), where ε is the singular perturbation parameter and γ is a positive constant. Our main effort is to make this γ smaller, say from 1/2 to 1/4 or less for Helmholtz type equations, by using appropriate coordinate stretching. Similar results are also obtained for advection-diffusion equations. Two important features of the proposed method are as follows: (a) the coordinate transformation does not involve the singular perturbation parameter ε; (b) machine accuracy can be achieved with N of the order of several hundreds, even when ε is very small. This is in contrast with conventional spectral, finite difference or finite element methods.