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Preconditioned Implicit Solvers for Nonlinear PDEs in Monument Conservation

Mathematical models for the description, in a quantitative way, of the damages induced on monuments by the action of specific pollutants are often systems of nonlinear, possibly degenerate, parabolic equations. Although some of the asymptotic properties of the solutions are known, one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest. In this paper an implicit numerical method is proposed, analyzed, and numerically tested for parabolic equations of porous media type and on a systems of two PDEs that models the sulfation of marble in monuments. Due to the nonlinearity inside the differential operator of the underlying mathematical model, the use of an iterative solver for a large system of nonlinear equations is required, and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures. Numerical experiments for the validation of the analysis complement this contribution.