Krylov methods in the finite element solution of groundwater transport problems

The numerical treatment of the groundwater transport model requires the repeated solution of nonsymmetric, large, sparse systems of linear equations. For this reason it is important to employ algorithms that are efficient and robust in terms of both memory utilization and computational burden. Among the most efficient schemes, we can mention those based on the definition of appropriate Krylov subspaces (Conjugate Gradient like) and those based on the Lanczos algorithm (Lanczos like). These two classes are often intersected, and mixed techniques seems to be the most attractive in terms of efficiency and robustness. In the solution of the groundwater transport equation, these schemes may perform poorly, or even may not converge at all, in advection dominated cases. In this paper we compare the efficiency of some of these algorithms recently proposed in the literature, applied to the solution of nonsymmetric systems arising from the finite element discretization of the groundwater transport equation. We consider Saad`s GMRES, Freund's TFQMR, and van der Vorst BiCGSTAB. We also investigate the effects of different preconditioning strategies: the diagonal scaling, ILU, and the recently proposed ILUT preconditioner, an implementation of the incomplete factorization that allows for variable fill inn of the triangular factors. The various schemes are compared on the basis of their performance in two sample problems, with constant and variable velocity fields.