An optimal rational basis function in the finite element method for convection-diffusion problems in one space variables

Abstract

A new fixed grid Galerkin finite element technique using a rational basis function of order (1, 1) to solve convection-dominated phenomenon is described. A special feature of the method is that it employs rational basis functions, which are biased in the upstream direction and thus replace the equivalent upwinding techniques. Optimality of the rational basis function results in an accurate and stable numerical scheme which is used to avoid numerical oscillations at high Peclet numbers. Numerical examples are discussed which illustrate the efficiency of the method.