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

Abstract

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 $L_2$-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.