Preconditioned fully implicit PDE solvers for monument conservation

TL;DR

This paper develops and tests a fully implicit numerical method for simulating damage in monuments caused by pollutants, focusing on nonlinear parabolic PDEs related to marble sulfation, with emphasis on efficient solvers.

Contribution

It introduces a novel fully implicit scheme for nonlinear PDEs in monument conservation, with spectral analysis and advanced iterative solvers for large linear systems.

Findings

The method accurately models sulfation damage in marble monuments.

Preconditioned Krylov and multigrid solvers improve computational efficiency.

Numerical experiments validate the effectiveness of the proposed approach.

Abstract

Mathematical models for the description, in a quantitative way, of the damages induced on the monuments by the action of specific pollutants are often systems of nonlinear, possibly degenerate, parabolic equations. Although some the asymptotic properties of the solutions are known, for a short window of time, one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest. In this paper a fully 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 nonlinear nature of the underlying mathematical model, the use of a fixed point scheme 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 or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures. Numerical experiments for the validation of the analysis complement this contribution.