The error estimates of spectral methods for 1-dimension singularly perturbed problem

Abstract

In this paper, we study the a posteriori error estimates of Galerkin spectral methods for the singularly perturbed problem on a unit interval. By the generalized orthogonal Jacobi polynomials, the lower bound of orthogonal coefficients, and the upper bound of the a posteriori error estimates are rigorously proposed. Our estimates consist of the truncation error of weighted L2-projection of the right hand function and limit low frequency terms of the coefficients within Jacobi polynomial expansions. While the low frequency coefficients are together with exponentially decaying multiples, the decaying rate of high frequency coefficients of the right hand function can be used to measure the approximation accuracy of the numerical solution.