A discontinuous Galerkin method for the Cahn-Hilliard equation

Choo, Sang-Mok; Lee, Y. J.

doi:10.1007/bf02936559

Cited by 25 publications

(21 citation statements)

References 7 publications

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“…There have been many algorithms developed and simulations performed for the C-H equations, using Finite Element Methods [18][19][20][21][22][23][24][25], Discontinuous Galerkin Techniques [26][27][28], Finite Difference Schemes [29][30][31][32][33][34][35][36], Spectral Methods [37,38], Collocation Techniques [39][40][41], Adomian Decomposition Procedure [42], m-transform [43] and etc.…”

Section: Introductionmentioning

confidence: 99%

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

Dehghan

Mirzaei

2009

Engineering Analysis with Boundary Elements

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Section: Introductionmentioning

confidence: 99%

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

Dehghan

Mirzaei

2009

Engineering Analysis with Boundary Elements

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“…The papers of Babuška and Zlámal [4] and Baker [6] are the earliest contributions to the theory of discontinuous Galerkin finite element methods for fourth-order elliptic problems; for more recent results, including historical notes, see [47,34,49,50,48]. The application of discontinuous Galerkin methods to the Cahn-Hilliard equation is discussed in [20,56,58,35]. In particular, in the article of Feng and Karakashian [35] a fully-discrete discontinuous Galerkin method is analyzed for the Cahn-Hilliard equation written as a fourth-order PDE, and an optimal-order error bound is derived for the order-parameter c in the broken L 2 (H 2 ) norm with discontinuous piecewise polynomials of degree p ≥ 2.…”

mentioning

confidence: 99%

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

Kay¹,

Styles²,

Süli³

2009

SIAM J. Numer. Anal.

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Abstract. The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn-Hilliard equation with convection. Using discontinuous piecewise polynomials of degree p ≥ 1 and backward Euler discretization in time, we show that the order-parameter c is approximated in the broken L ∞ (H 1 ) norm with optimal order, O(h p + τ ); the associated chemical potential w = Φ (c) − γ 2 ∆c is shown to be approximated with optimal order, O(h p + τ ), in the broken L 2 (H 1 ) norm. Here Φ(c) = 1 4(1 − c 2 ) 2 is a quartic free-energy function and γ > 0 is an interface parameter. Numerical results are presented with polynomials of degree p = 1, 2, 3.

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“…6 is solved using the proposed finite volume method [37]. The CahnHilliard equations have been solved using finite elements [18, 19, 45-47, 61, 62], finite differences [50,91] and, more recently, discontinuous Galerkin methods [30,104,106]. We solve the second example in [104].…”

Section: Cahn-hilliard Equationmentioning

confidence: 99%

High-order Finite Volume Methods and Multiresolution Reproducing Kernels

Cueto‐Felgueroso

Colominas

2008

Arch Computat Methods Eng

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This paper presents a review of some of the most successful higher-order numerical schemes for the compressible Navier-Stokes equations on unstructured grids. A suitable candidate scheme would need to be able to handle potentially discontinuous flows, arising from the predominantly hyperbolic character of the equations, and at the same time be well suited for elliptic problems, in order to deal with the viscous terms. Within this context, we explore the performance of Moving Least-Squares (MLS) approximations in the construction of higher order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: 1) computation of high order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, 2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and 3) multiresolution shock detection and selective limiting. The proposed finite volume method is formulated within a continuous spatial representation framework, provided by the MLS approximants, which is "broken" locally (inside each cell) into piecewise polynomial expansions, in order to make use of the specialized finite volume technology for hyperbolic problems. This approach is in contrast with the usual practice in the finite volume literature, which proceeds bottomup, starting from a piecewise constant spatial representa-L. Cueto-Felgueroso ( ) tion. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest, and very competitive when compared to other existing procedures.

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A discontinuous Galerkin method for the Cahn-Hilliard equation

Cited by 25 publications

References 7 publications

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation

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

High-order Finite Volume Methods and Multiresolution Reproducing Kernels

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