Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transition

Zee, Kristoffer G. van der; Oden, J. T.; Prudhomme, Serge; Hawkins-Daarud, Andrea

doi:10.1002/num.20638

Cited by 33 publications

(23 citation statements)

References 70 publications

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“…Oden & S. Prudhomme [38,25], who introduced the term goal oriented adaptivity in this context. We can find applications in structural problems and visco-elasticity [27,48,22,21], in fluidstructure interactions [46,45,47], and in control theory [15,17,16]. The goal oriented adaptivity is also a key feature for some inverse problems, like the determination of the composition of the underground.…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

Darrigrand

Pardo

Muga

2015

Computers & Mathematics with Applications

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In this work, the error of a given output functional is represented using bilinear forms that are different from those given by the adjoint problem. These representations can be employed to design novel h, p, and hp energy-norm and goal-oriented adaptive algorithms. Numerical results in 1D show that, for wave propagation problems, the advantages of this new representation are notorious when selecting the Laplace equation as the dual problem. Specifically, the upper bounds of the new error representation are sharper than the classical ones used in both energy-norm and goal-oriented adaptive methods, especially when the dispersion (pollution) error is significant.

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Section: Introductionmentioning

confidence: 99%

Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

Darrigrand

Pardo

Muga

2015

Computers & Mathematics with Applications

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“…There exist numerous engineering applications that motivate the use of GOA, including electromagnetics, 4-8 structural problems and visco-elasticity, [9][10][11][12][13] fluid-structure interactions, [14][15][16] and control theory. [17][18][19] Apart from these applications, convergence properties of GOA have also been recently studied in Pollock, Holst et al, Holst and Pollock, Mommer and Stevenson, and Feischl et al [20][21][22][23][24] The origin of the GOA is in the works of Rannacher et al [25][26][27] followed by the works of Peraire; Patera et al [28][29][30][31][32][33] on a posteriori error estimates of the error in the QoI.…”

Section: Introductionmentioning

confidence: 99%

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Darrigrand

Rodríguez-Rozas

Muga

et al. 2017

Numerical Meth Engineering

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SummaryIn goal-oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal-oriented adaptivity.While the method can be applied to a variety of problems, we focus here on two-and three-dimensional (2-D and 3-D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.

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“…Over the years, they have been developed for time-independent, time-dependent, linear, nonlinear and coupled problems; see e.g., [22,41,40,11,39,42]. Goal-oriented estimates are explicit dual-based estimates, as they directly compute an approximation to the dual problem, in contrast to the above mentioned duality techniques (where they are only used as auxiliary problems for deriving estimates).…”

Section: Introductionmentioning

confidence: 99%

“…Here, C 0 (u; w, v) denotes the Gâteaux (or Fréchet) derivative of C at u in the direction of w [6,19,39]:…”

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confidence: 99%

Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

Şimşek

Zee

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator.

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Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transition

Cited by 33 publications

References 70 publications

Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

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