AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation

Donatelli, Marco; Semplice, Matteo; Serra‐Capizzano, Stefano

doi:10.1016/j.apnum.2013.02.001

Cited by 7 publications

(11 citation statements)

References 15 publications

(44 reference statements)

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“…Grid setup This case can be efficiently dealt with spatial discretizations based on a staggered grid arrangement of the variables; in particular, s variables are stored at the nodes of a Cartesian grid, while c variables are stored on the dual grid. The main grid can be uniform as in [22] or non uniform as in [13] and is illustrated in Fig. 1b.…”

Section: Numerical Approximation For Cartesian Domainsmentioning

confidence: 99%

“…In this work we focus on the numerical solution of the model of [2], recalled in §2, revising previous work of the authors on this subject. As a first step beyond the onedimensional discretizations two-dimensional Cartesian domains were considered in [22] and a better work/precision efficiency was obtained in [13] with the use of non-uniform Cartesian discretizations. These are described in §3.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Simulations of Marble Sulfation

Coco

Donatelli

Semplice

et al. 2020

Mathematical Modeling in Cultural Heritage

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In this chapter we describe some computational techniques to approximate the evolution of gypsum crusts on marble monuments. Mathematical models of this phenomenon are typically based on partial differential equations and here we deliberately consider a quite simple one, so that we can focus on the numerical techniques that can be used to overcome the main difficulties of this kind of computations, namely the efficiency of the timestepping procedure and the complexity of the computational domains in real-world cases. First, the design of optimal preconditioners for Cartesian grid discretizations is reviewed. Then, we illustrate a technique to deal with non Cartesian domains by described via a level-set function. The chapter ends with a study of the influence of the surface curvature on the growth of the gypsum crust, that generalizes earlier analytical one-dimensional results.

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Section: Numerical Approximation For Cartesian Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of Marble Sulfation

Coco

Donatelli

Semplice

et al. 2020

Mathematical Modeling in Cultural Heritage

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“…( 15),( 21)), and ∆s * and ∆c * are defined as in (22). If we choose τ s and τ c as in (24) to have a Gauss-Seidel iteration, i.e.…”

Section: Relaxation Operatormentioning

confidence: 99%

“…Within this paper we consider a novel numerical technique for the approximation of nonlinear (possibly degenerate) parabolic equations, which relies on the finite difference discretization and efficient solvers of [49,23,24] and on the level-set domain description and handling of boundary conditions of [18]. As in [49], the time discretization is the implicit Crank-Nicolson, a large nonlinear system at each time step is solved by a Newton-Raphson procedure, with a tailored multigrid technique for the linear systems.…”

Section: Introductionmentioning

confidence: 99%

A level-set multigrid technique for nonlinear diffusion in the numerical simulation of marble degradation under chemical pollutants

Coco,

Semplice,

Serra-Capizzano

2019

Preprint

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Having in mind the modelling of marble degradation under chemical pollutants, e.g. the sulfation process, we consider governing nonlinear diffusion equations and their numerical approximation. The space domain of a computation is the pristine marble object and, in order to accurately discretize it while maintaining the simplicity of finite difference discretizations, the domain is implicitly defined using a level-set approach. A uniform Cartesian grid is laid over a box containing the domain, but the solution is defined and updated only in the grid nodes that lie inside the domain, the level-set being employed to select them and to impose accurately the boundary conditions. We use a Crank-Nicolson in time, while for the space variables the discretization is performed by a standard Finite-Difference scheme for grid points inside the domain and by a ghost-cell technique on the ghost points (by using boundary conditions). The solution of the large nonlinear system is obtained by a Newton-Raphson procedure and a tailored multigrid technique is developed for the inner linear solvers. The numerical results, which are very satisfactory in terms of reconstruction quality and of computational efficiency, are presented and discussed at the end of the paper.

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“…[4,17,18,21,27,42] amongst others). There exists very little convergence theory for the nonlinear methods (e.g.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient

Brabazon

Hubbard

Jimack

2014

Computers & Mathematics with Applications

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. AbstractNonlinear multigrid methods such as the Full Approximation Scheme (FAS) and Newton-multigrid (Newton-MG) are well established as fast solvers for nonlinear PDEs of elliptic and parabolic type. In this paper we consider Newton-MG and FAS iterations applied to second order differential operators with nonlinear diffusion coefficient. Under mild assumptions arising in practical applications, an approximation (shown to be sharp) of the execution time of the algorithms is derived, which demonstrates that Newton-MG can be expected to be a faster iteration than a standard FAS iteration for a finite element discretisation. Results are provided for elliptic and parabolic problems, demonstrating a faster execution time as well as greater stability of the Newton-MG iteration. Results are explained using current theory for the convergence of multigrid methods, giving a qualitative insight into how the nonlinear multigrid methods can be expected to perform in practice.

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AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation

Cited by 7 publications

References 15 publications

Numerical Simulations of Marble Sulfation

Numerical Simulations of Marble Sulfation

A level-set multigrid technique for nonlinear diffusion in the numerical simulation of marble degradation under chemical pollutants

Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient

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