Error Analysis for Boundary Layer Resolving Galerkin-spectral Method

In this paper, we investigate a Galerkin-spectral method, which employs coordinate stretching and a new class of trail functions, for solving the singularly perturbed boundary value problems. An error analysis for the proposed spectral method will be presented. Two transformation functions are considered in detail. We show that for Helmholtz type equations spectral accuracy can be obtained with these two transformations provided that N = O(?), where N is the total number of grid points, is the perturbation parameter , is a positive constant satisfying 1=4. The constant is adjustable and can be made small by changing a free parameter in the transformation functions. Similar results are also obtained for advection-diiusion equations. Numerical experiments connrm that our method is very eecient for solving singularly perturbed problems with boundary layers. Two important features of the proposed method are as follows: (a) The choice of grids used for the proposed method is independent of the width of the boundary layers. It is to be shown in this paper that many eecient methods proposed before require the information of the layer width. In other words, a successful grid for certain type of boundary layer problems (say Helmholtz type equations) may not work for another type of problems (say advection-diiusion equations). Our method does not require the information for the width, yet can resolve very thin boundary layers. (b) Spectral accuracy can be recovered for N at most of order O(100), even when is very small. This is in contrast with conventional spectral, nite diierence or nite element methods.