Multigrid
A Globally Convergent Newton-GMRES Subspace Method for Systems of Nonlinear Equations
SIAM Journal on Scientific Computing
The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences
Journal of Approximation Theory
On the Asymptotic Spectrum of Finite Element Matrix Sequences
SIAM Journal on Numerical Analysis
Preconditioned Implicit Solvers for Nonlinear PDEs in Monument Conservation
SIAM Journal on Scientific Computing
Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations
SIAM Journal on Matrix Analysis and Applications
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Motivated by the modelling of marble degradation by chemical pollutants, we consider the approximation by implicit finite differences schemes of nonlinear degenerate parabolic equations in which sharp boundary layers naturally occur. The latter suggests to consider various types of nonuniform griddings, when defining suitable approximation schemes. The resulting large nonlinear systems are treated by Newton methods, while the locally Toeplitz linear systems arising from the Jacobian have to be solved efficiently. To this end, we propose the use of AMG preconditioners and we study the related convergence issues, together with the associated spectral features. We present some numerical experiments supporting our theoretical results on the spectrum of the coefficient matrix of the linear systems, alongside others regarding the numerical simulations in the case of the specific model.