Krylov Methods for Large Sparse Systems: A Comprehensive Overview

In this paper are analyzed behavior and properties for different Krylov methods applied in different categories of problems. These categories often include PDEs, econometrics and network models, which are represented by large sparse systems. For our empirical analysis are taken into consideration size, the density of non-zero elements, symmetry/un-symmetry, eigenvalue distribution, also well/ill-conditioned and random systems. Convergence, approximation error and residuals are compared for the full version of methods, some restarted methods and preconditioned methods. Two preconditioners are considered respectively, ILU(0) and IC(0) by using at least five preconditioning techniques. In each case, empirical results show which technique is best to use based on properties of the system and are backed up by general theoretical information already found on Krylov space methods.

Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

This work deals with the efficient iterative solution of the system of equations stemming from mimetic finite difference discretization of a hybrid-dimensional mixed Darcy problem modeling flow in fractured porous media. We investigate the spectral properties of a mixed discrete formulation based on mimetic finite differences for flow in the bulk matrix and finite volumes for the fractures, and present an approximation of the factors in a set of approximate block factorization preconditioners that accelerates convergence of iterative solvers applied to the resulting discrete system. Numerical tests on significant three-dimensional cases have assessed the properties of the proposed preconditioners.

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.

Optimized chemical preconditioning of Philippine natural zeolites

Natural zeolites in the Aksitero sedimentary formation of the western Luzon area of the Philippines were evaluated. The natural washed zeolite (NW-Z) was preconditioned with acid to purify it and increase its surface area. Acid treatment with 3 M HCl for 12 h yielded optimum acid treatment of the NW-Z, causing increases in the Si/Al ratio, the specific surface area by 32.5% and the porosity of the acid-treated zeolite (HC-Z). The HC-Z was washed with 4 M NaCl for 3, 6, 12, 18, 24, 48 and 72 h to improve its cation-exchange capacity for copper. The sodium-treated zeolite (Na-Z) was immersed in 100 ppm CuSO4 solution to test the copper-uptake capacity. Pretreatment of HC-Z with 4 M NaCl for 24 h is optimal for sodium treatment of the preconditioned HC-Z. The preconditioning techniques did not significantly alter the structure and morphology of the zeolite samples. It is suggested that the preconditioned Philippine natural zeolite samples are readily available for further functionalization that will enhance their antibacterial, catalytic and adsorption properties, with various useful applications in the field of catalysis, biomedicine, environmental mitigation and wastewater treatment.