Abstract
A preconditioned iterative solution method is presented for nonlinear parabolic transport systems. The ingredients are implicit Euler discretization in time and finite element discretization in space, then an outer-inner (outer damped inexact Newton method with inner preconditioned conjugate gradient) iteration, further, as a main part, preconditioning via an l-tuple of independent elliptic operators. Numerical results show that the suggested method works properly for a test problem in air pollution modeling.
Highlights
Nonlinear time-dependent reaction-convection-diffusion systems arise in various situations in applied mathematics and mathematical modeling, often leading to large-scale, computationally challenging problems
A further specific feature of air pollution models is that chemical reactions are described by very stiff systems of ODEs, which makes the system ill-conditioned
Following [1], we propose an outer-inner iteration for solving the finite element discretization of the nonlinear elliptic problems
Summary
Nonlinear time-dependent reaction-convection-diffusion (transport) systems arise in various situations in applied mathematics and mathematical modeling, often leading to large-scale, computationally challenging problems. Ui ∂Ω×R+ = γi, which contains nonlinear coupling in the reaction terms Such problems frequently arise in environmental modeling, for instance in the study of the transport of air pollutants, where ui are concentrations of chemical species. The main part of this method is preconditioning using the discretization of an -tuple of independent elliptic operators as preconditioner This implies that the preconditioning matrix has a block-diagonal structure, and the auxiliary problems can be solved with a cost proportional to that of a single PDE, in contrast to solving the linearized PDE systems. We note that the applicability of the proposed method is more general than the mentioned air pollution models This approach can be used in a similar way for other time-dependent parabolic reaction-convection-diffusion systems arising in the context of chemical or biological interactions. Preconditioned Iterative Solution of Nonlinear Parabolic Systems 643 for this realistic model problem
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