An approximation technique for solutions of singularly perturbed one-dimensional convection-diffusion problem

In this study, a weighted residual method is presented in order to numerically solve singularly perturbed one-dimensional parabolic convection-diffusion problem. Assuming an approximate polynomial solution of a prescribed degree N, the method uses the set of bivariate monomials whose degrees do not exceed N as the set of base functions. Then, following Galerkin's path, inner product with the base functions are applied to the residual of the approximate solution polynomial. Incorporation of the initial and boundary conditions are ensured by forcing the approximate solution to satisfy these conditions on equidistant collocation points. The solution of the resulting linear system then yields the approximate polynomial solution. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed briefly. The Galerkin-like scheme and the residual correction technique are illustrated with two examples. The obtained results are also compared with other methods presented in the literature.