Keywords: incomplete factorizations, preconditioners, linear systems, environmental modelling, convection-diffusion, active carbon filters.

Many industrial or natural processes of environmental interest are transient convection-diffusion-reaction problems. This is the case, for instance, of the operation of activated-carbon filters and the dispersion of pollutants in the atmosphere. In these technological applications, there is a growing demand for 3D simulations. Due to the transient, 3D nature of the problems, efficient finite element models are needed.

With computational efficiency in mind, we have chosen to stabilize the convective term with a lest-squares technique. This results in a symmetric positive definite (SPD) system [1] to be solved at each time-step.

We solve these linear systems by means of the preconditioned conjugate gradient method. Incomplete Cholesky factorizations [2] are used as preconditioners. We have compared the numerical performance of two different strategies: threshold incomplete factorizations based on a drop tolerance [3] and incomplete factorizations with prescribed density [4]. To put the results in context, both a complete Cholesky factorization (i.e. direct solver) and a diagonal Jacobi preconditioner have also been used.

Our numerical experiments show that drop-tolerance factorizations turn out to be more computationally efficient --in terms of CPU time and non-zero entries-- than prescribed-memory factorizations. However, since memory requirements cannot be predicted a priori in threshold factorizations, the latter approach may be preferable when memory is a critical issue.

Experiments also show that, for transient convection-diffusion problems, the numerical efficiency of a preconditioner is completely controlled by the application stage at each time-step. Since the computational cost of the computation stage of the preconditioner can be amortized over many time-steps (tens of thousands in our applications), how much it costs to obtain the factorization turns out to be an irrelevant factor.

1

J. Donea, A. Huerta, "Finite element methods for flow problems", John Wiley & Sons, Chichester, 2003.

2

Y. Saad, "Iterative methods for sparse linear systems", Society for Industrial and Applied Mathematics, Philadelphia, 2003.

3

N. Munksgaard, "Solving sparse symmetric sets of linear equations by preconditioned conjugate gradients", ACM Trans. Math. Softw., 6:206-219, 1980. doi:10.1145/355887.355893

4

C-J Lin, J.J. Moré, "Incomplete Cholesky factorizations with limited memory", SIAM J. Sci. Comput., 21(1):24-45, 1999. doi:10.1137/S1064827597327334

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