A novel exponentially fitted triangular finite element method for an advection-diffusion problem with boundary layers

In this paper we develop an exponentially fitted finite element method for a singularly perturbed advection-diffusion problem with a singular perturbation parameter epsilon. This finite element method is based on a set of novel piecewise exponential basis functions constructed on unstructured triangular meshes, The basis functions can not be expressed explicitly, but the values of each of them and its associated flux at a point are determined by a set of two-point boundary value problems which can be solved exactly. A method for evaluating elements of the stiffness matrix is also proposed for the case that a is small. Numerical results, presented to validate the method, show that the method is stable for a large range of epsilon. It is also shown by the numerical results that the rate of convergence of the method in an energy norm is of order h(1/2) when epsilon is small. (C) 1997 Academic Press.