Preconditioned Iterative Method for Reactive Transport with Sorption in Porous Media

This work deals with the numerical solution of a nonlinear degenerate parabolic equation arising in a model of reactive solute transport in porous media, including equilibrium sorption. The model is a simplified, yet representative, version of multicomponents reactive transport models. The numerical scheme is based on an operator splitting method, the advection and diffusion operators are solved separately using the upwind finite volume method and the mixed finite element method (MFEM) respectively. The discrete nonlinear system is solved by the Newton-Krylov method, where the linear system at each Newton step is itself solved by a Krylov type method, avoiding the storage of the full Jacobian matrix. A critical aspect of the method is an efficient matrix-free preconditioner. Our aim is, on the one hand to analyze the convergence of fixed-point algorithms. On the other hand we introduce preconditioning techniques for this system, respecting its block structure then we propose an alternative formulation based on the elimination of one of the unknowns. In both cases, we prove that the eigenvalues of the preconditioned Jacobian matrices are bounded independently of the mesh size, so that the number of outer Newton iterations, as well as the number of inner GMRES iterations, are independent of the mesh size. These results are illustrated by some numerical experiments comparing the performance of the methods.