A new efficient spectral Galerkin method for singular perturbation problems

A new spectral Galerkin method is proposed for the convection-dominated convection-diffusion equation. This method employs a new class of trail function spaces. The available error bounds provide a clear theoretical interpretation for the higher accuracy of the new method compared to the conventional spectral methods when applied to problems with thin boundary layers. Efficient solution techniques are developed for the convection-diffusion equations by using appropriate basis functions for the new trial function spaces. The higher accuracy and the effectiveness of the new method for problems with thin boundary layers are confirmed by our numerical experiments.