<i><b>A posteriori</b></i>error control for the Allen–Cahn problem: circumventing Gronwall's inequality

Kessler, Daniel; Nochetto, Ricardo H.; Schmidt, Alfred

doi:10.1051/m2an:2004006

Cited by 108 publications

(100 citation statements)

References 10 publications

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“…For nonlinear time-dependent problems, there are two commonly used approaches for deriving conditional a posteriori error bounds: continuation arguments, cf. [5,14], and fixed point arguments, cf. [6,15].…”

Section: U(t) H < ∞ For 0 < T < T ∞ Lim T T ∞ U(t) H = ∞mentioning

confidence: 99%

hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

2017

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This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated.

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Section: U(t) H < ∞ For 0 < T < T ∞ Lim T T ∞ U(t) H = ∞mentioning

confidence: 99%

hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

2017

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“…For residual-based estimates, the difficulty is the derivation of sharp bounds by eliminating the exponential dependence on the interface thickness; see, e.g., (Kessler et al, 2004;Feng and Wu, 2005;Bartels, 2005) for Allen-Cahn estimates and (Feng and Wu, 2008;Bartels and Müller, 2010) for Cahn-Hilliard estimates. Alternatively, estimates based on the computation of a dual problem have recently been developed (Simsek et al, 2015) in the spirit of duality-based estimates for nonlinear parabolic problems (Eriksson et al, 2004).…”

Section: Adaptive Mesh and Time-step Refinementmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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“…Explicit Euler discretizations for Allen-Cahn obstacle problems have been used for example in [12,20,23,24]. Numerical analysis for (semi-) implicit discretizations of the Allen-Cahn model has been performed in the papers [15,22,31,32,33,34] and in works cited in these papers. Fully implicit discretizations are the most accurate, see e.g.…”

Section: Primal-dual Active Set Approachmentioning

confidence: 99%

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

Blank

Garcke

Sarbu

et al. 2013

IMA Journal of Numerical Analysis

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We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency.

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A posteriorierror control for the Allen–Cahn problem: circumventing Gronwall's inequality

Cited by 108 publications

References 10 publications

hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

hp-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

Computational Phase‐Field Modeling

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

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