Non-oscillatory and Non-diffusive Solution of Convection Problems by the Iteratively Reweighted Least-Squares Finite Element Method

Abstract

In this paper we first introduce the least-squares finite element method (LSFEM) for two-dimensional steady-state pure convection problems with smooth solutions and compare the LSFEM with other finite element methods. We prove that the LSFEM has the same stability estimate as the original equation; i.e., the LSFEM has better control of the streamline derivative than the streamline upwinding Petrov-Galerkin method. Numerical convergence rates are given to show that the LSFEM is almost optimal. Then we use this LSFEM as a framework to develop an iteratively reweighted least-squares finite element method (IRLSFEM) to obtain non-oscillatory and non-diffusive solutions for problems with contact discontinuities. This new method produces a highly accurate numerical solution that has a sharp discontinuity in one element. A number of examples solved by using triangular and bilinear elements are presented to show that the method can convect contact discontinuities without error.