The finite element method with weighted basis functions for singularly perturbed convection–diffusion problems

In this paper, we present a finite element method for singularly perturbed convection–diffusion problems in both one and two dimensions, based on a set of weighted basis functions constructed on unstructured meshes (in 2D). For the one-dimensional case, both first and second-order schemes are discussed. A technique for approximating fluxes is proposed. Some theoretical results on uniform convergence are obtained. For the two-dimensional case, a first-order scheme is constructed for problems with two singular perturbation parameters. A technique is also developed in approximating fluxes in 2D. This technique is used to simplify the calculation of the integrals in the stiffness matrix arising from the scheme, which will save computational costs. The numerical results support the theoretical results and demonstrate that the method is stable for a wide range of singular perturbation parameters.