TAYLOR-LEAST-SQUARES FINITE-ELEMENT FOR 2-DIMENSIONAL ADVECTION-DOMINATED UNSTEADY ADVECTION DIFFUSION-PROBLEMS

A new finite element method, the Taylor–least‐squares, is proposed to approximate the advection‐dominated unsteady advection–diffusion equation. The new scheme is a direct generalization of the Taylor–Galerkin and least‐squares finite element methods. Higher‐order spatial derivatives in the new formulation necessitate higher‐degree polynomials. Hermite cubic shape functions are used. Extensive comparisons with other methods in one dimension proved that the new scheme is a step forward in modelling this difficult problem. The method offers straightforward generalizations to higher dimensions without losing the accuracy demonstrated in one dimension, i.e. the method preserves the important property of the Taylor–Galerkin scheme of being free of numerical crosswind diffusion. Several numerical experiments were made in two dimensions and excellent results were obtained from the representative experiments. Copyright © 1990 John Wiley & Sons, Ltd