Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection

Kay, David; Styles, Vanessa; Süli, Endre

doi:10.1137/080726768

Cited by 74 publications

(67 citation statements)

References 55 publications

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“…As no analytical solution is available, a manufactured solution [27,30] is utilised. The general idea is to use an arbitrary function as analytic reference solution.…”

Section: Convergence Analysis Based On a Manufactured Solutionmentioning

confidence: 99%

“…Zhang et al [27] have presented a convergence analysis based on the method of manufactured solutions [28]. By assuming an appropriate solution and extending the underlying differential equation by a source term, different discretisations of the same phase-field model can be compared, see also Aristotelous et al [29] and Kay et al [30]. To the authors' knowledge a quantitative convergence study for diffuse interface models has never been carried out within the framework of isogeometric analysis.…”

Section: Introductionmentioning

confidence: 99%

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Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Kästner

Metsch

Borst

2016

Journal of Computational Physics

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Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. AbstractHerein, we present a numerical convergence study of the Cahn-Hilliard phase-field model within an isogeometric finite element analysis framework. Using a manufactured solution, a mixed formulation of the Cahn-Hilliard equation and the direct discretisation of the weak form, which requires a C 1 -continuous approximation, are compared in terms of convergence rates. For approximations that are higher than second-order in space, the direct discretisation is found to be superior. Suboptimal convergence rates occur when splines of order p = 2 are used. This is validated with a priori error estimates for linear problems. The convergence analysis is completed with an investigation of the temporal discretisation. Second-order accuracy is found for the generalised-α method. This ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the Cahn-Hilliard equation. The isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition.

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“…As no analytical solution is available, a manufactured solution [27,30] is utilised. The general idea is to use an arbitrary function as analytic reference solution.…”

Section: Convergence Analysis Based On a Manufactured Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Kästner

Metsch

Borst

2016

Journal of Computational Physics

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“…Therefore this bubble forms a boomerang-shaped phase island. These results correspond to the computations of Kay et al [36] using a discontinuous Galerkin discretization for a similar problem.…”

Section: Phase Decomposition With Convectionmentioning

confidence: 82%

“…Standard finite element basis functions provide C 0 -continuity only, and extending the finite element space by Hermitian polynomials is elaborate and restricted to rectangular domains [50]. An alternative approach is a mixed finite element formulation, applied, e.g., in [36,48,49]. However, it is numerically expensive because it introduces additional unknowns to the primal degrees of freedom.…”

Section: Phase Separation In a Flow Fieldmentioning

confidence: 99%

A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation

Anders

Weinberg

2011

Z Angew Math Mech

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The mixing and de-mixing properties of highly viscous fluids can be described as a diffusive system driven by an external velocity field. In order to analyze the micro-morphological evolution with a diffusion theory of heterogeneous mixtures we apply a Cahn-Hilliard phase-field model, which is here extended by a convective term. As the model is based on an energetic variational formulation, we state the underlying energy minimizing principles. For consistent finite element analysis we provide a piecewise smooth and globally C 1 -continuous approximation by means of rational B-spline basis functions. Numerical examples exhibit the physical properties of the model. Simulations of phase decomposition and coarsening controlled by diffusion and by convection show the robustness and the versatility of our approach.

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“…From these properties is possible to deduce the existence and uniqueness of global in time weak solution (φ, w) of (P ), [1,8,28,31,46] and it is known that this system does not satisfy a maximum principle property: " |φ(t, x)| < 1 a.e. in Ω for each t ≥ 0 if the initial data satisfies |φ 0 (x)| < 1 a.e.…”

Section: Cahn-hilliard Equationmentioning

confidence: 99%

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Tierra

Guillén-González

2014

Arch Computat Methods Eng

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In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters.Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.

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Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection

Cited by 74 publications

References 55 publications

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

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