A Posteriori error estimates of the spectral method for singularly perturbed convection-diffusion problem

In this study, we present a posteriori error analysis for the singularly perturbed convection-diffusion equations with two parameters, addressing an approximation theory and numerical analysis. This paper develops a new spectral element method that crucially combines two orthogonal polynomials, shifted Legendre and Jacobi, with the two singular parameters. We derive an upper bound for the connection coefficients of the proposed spectral approximation, and it is further utilized to obtain a posteriori error bounds with respect to the two singular parameters. Test problems are utilized to substantiate the theoretical analysis. The effectiveness of the proposed method is studied by making a comparison with the other methods' solutions, and the quality of the solution is ensured through figures and tables.