Differential Equations and Nonlinear Mechanics
Volume 2008 (2008), Article ID 267454, 21 pages
Research Article

Bubble-Enriched Least-Squares Finite Element Method for Transient Advective Transport

Rajeev Kumar and Brian H. Dennis

Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX 76019, USA

Received 4 March 2008; Revised 9 July 2008; Accepted 5 September 2008

Academic Editor: Emmanuele Di Benedetto

Copyright © 2008 Rajeev Kumar and Brian H. Dennis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The least-squares finite element method (LSFEM) has received increasing attention in recent years due to advantages over the Galerkin finite element method (GFEM). The method leads to a minimization problem in the L2-norm and thus results in a symmetric and positive definite matrix, even for first-order differential equations. In addition, the method contains an implicit streamline upwinding mechanism that prevents the appearance of oscillations that are characteristic of the Galerkin method. Thus, the least-squares approach does not require explicit stabilization and the associated stabilization parameters required by the Galerkin method. A new approach, the bubble enriched least-squares finite element method (BELSFEM), is presented and compared with the classical LSFEM. The BELSFEM requires a space-time element formulation and employs bubble functions in space and time to increase the accuracy of the finite element solution without degrading computational performance. We apply the BELSFEM and classical least-squares finite element methods to benchmark problems for 1D and 2D linear transport. The accuracy and performance are compared.