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

## Abstract

*c*is approximated in the broken ${\rm L}^\infty({\rm H}^1)$ norm, with optimal order ${\cal O}(h^p+\tau)$; the associated chemical potential $w=\Phi'(c)-\gamma^2\Delta c$ is shown to be approximated, with optimal order ${\cal O}(h^p+\tau)$ in the broken ${\rm L}^2({\rm H}^1)$ norm. Here $\Phi(c)=\frac{1}{4}(1-c^2)^2$ is a quartic free-energy function and $\gamma>0$ is an interface parameter. Numerical results are presented with polynomials of degree $p=1,2,3$.

### MSC codes

### Keywords

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*J. Math. Anal. Appl.*, 61 (1977), pp. 713–734.

*SIAM J. Numer. Anal.*, 39 (2002), pp. 1749–1779.

*SIAM J. Numer. Anal.*, 19 (1982), pp. 742–760.

*SIAM J. Numer. Anal.*, 10 (1973), pp. 863–875.

*J. Comput. Phys.*, 190 (2003), pp. 371–397.

*Math. Comp.*, 31 (1977), pp. 45–59.

*IMA J. Numer. Anal.*, 16 (1996), pp. 257–287.

*Numer. Math.*, 88 (2001), pp. 255–297.

*IMA J. Numer. Anal.*, 16 (1996), pp. 111–139.

*Eur. J. Mech. B Fluids*, 23 (2004), pp. 759–780.

*Asymptot. Anal.*, 20 (1999), pp. 175–212.

*Comput. & Fluids*, 31 (2002), pp. 41–68.

*SIAM J. Numer. Anal.*, 41 (2003), pp. 306–324.

*Comput. Methods Appl. Mech. Engrg.*, 195 (2006), pp. 3293–3310.

*Math. Models Methods Appl. Sci.*, 14 (2004), pp. 1893–1903.

*Compact embeddings of broken Sobolev spaces and applications*, IMA J. Numer. Anal., to appear.

*J. Chem. Phys.*, 28 (1958), pp. 258–267.

*J. Stat. Phys.*, 77 (1994), pp. 183–197.

*Acta Metall. Mater.*, 9 (1961), pp. 795–801.

*J. Appl. Math. Comput.*, 18 (2005), pp. 113–126.

*The development of discontinuous Galerkin methods*, in Discontinuous Galerkin Methods (Newport, RI, 1999), Lect. Notes Comput. Sci. Eng. 11, Springer, Berlin, 2000, pp. 3–50.

*Math. Comp.*, 54 (1990), pp. 545–581.

*SIAM J. Numer. Anal.*, 35 (1998), pp. 2440–2463.

*ZAMM Z. Angew. Math. Mech.*, 83 (2003), pp. 731–754.

*Numer. Math.*, 63 (1992), pp. 39–65.

*Interior penalty procedures for elliptic and parabolic Galerkin methods*, in Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), Lecture Notes in Phys. 58, Springer, Berlin, 1976, pp. 207–216.

*SIAM J. Numer. Anal.*, 28 (1991), pp. 1310–1322.

*Numer. Math.*, 54 (1989), pp. 575–590.

*IMA J. Appl. Math.*, 39 (1987), pp. 97–128.

*SIAM J. Numer. Anal.*, 30 (1993), pp. 1622–1663.

*J. Differential Equations*, 128 (1996), pp. 387–414.

*The Cahn–Hilliard model for the kinetics of phase separation*, in Mathematical Models for Phase Change Problems, Internat. Ser. Numer. Math. 88, J. F. Rodrigues, ed., Birkhäuser-Verlag, Basel, 1989, pp. 35–73.

*J. Comput. Appl. Math.*, 206 (2007), pp. 843–872.

*J. Sci. Comput.*, 22/23 (2005), pp. 289–314.

*Math. Comp.*, 76 (2007), pp. 1093–1117.

*Interfaces Free Bound.*, 7 (2005), pp. 1–28.

*SIAM J. Numer. Anal.*, 39 (2002), pp. 2133–2163.

*J. Comput. Phys.*, 155 (1999), pp. 96–127.

*Comput. Methods Appl. Mech. Engrg.*, 45 (1986), pp. 285–312.

*J. Comput. Phys.*, 193 (2004), pp. 357–379.

*Modelling and Simulation of Multi-Component, Multi-Phase Fluid Flows*, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2002.

*Appl. Math. Comput.*, 160 (2005), pp. 589–606.

*Poincaré-type Inequalities for Broken Sobolev Spaces*, Numerical Analysis Technical report 3, Oxford University Computing Laboratory, Oxford, 2003.

*On a finite element method for solving the neutron transport equation*, in Mathematical Aspects of the Finite Element Method in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 89–123.

*Arch. Rat. Mech. Anal.*, 98 (1987), pp. 123–142.

*Comput. Methods Appl. Math.*, 7 (2007), pp. 365–375.

*Commun. Numer. Methods Engrg.*, 23 (2006), pp. 447–459.

*J. Sci. Comput.*, 30 (2007), pp. 465–491.

*Comput. Methods Appl. Math.*, 3 (2003), pp. 596–607.

*Comput. Methods Appl. Mech. Engrg.*, 196 (2007), pp. 1851–1863.

*Phys. Rev. E*, 75 (2007), p. 016216.

*SIAM J. Numer. Anal.*, 15 (1978), pp. 705–714.

*Triangular Mesh Methods for Neutron Transport Equation*, Technical report LA-UR-73-479, Los Alamos Scientific Library, Los Alamos, NM, 1973.

*SIAM J. Numer. Anal.*, 39 (2001), pp. 902–931.

*J. Comput. Phys.*, 205 (2005), pp. 401–407.

*J. Comput. Phys.*, 218 (2006), pp. 860–877.

*SIAM J. Numer. Anal.*, 15 (1978), pp. 152–161.

*J. Comput. Phys.*, 227 (2007), pp. 472–491.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 10 June 2008

**Accepted**: 26 February 2009

**Published online**: 31 July 2009

#### MSC codes

#### Keywords

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- A rotational pressure-correction discontinuous Galerkin scheme for the Cahn-Hilliard-Darcy-Stokes systemAdvances in Computational Mathematics, Vol. 50, No. 3 | 30 May 2024
- Energy-stable and boundedness preserving numerical schemes for the Cahn-Hilliard equation with degenerate mobilityApplied Numerical Mathematics, Vol. 196 | 1 Feb 2024
- Convergence of a Decoupled Splitting Scheme for the Cahn–Hilliard–Navier–Stokes SystemSIAM Journal on Numerical Analysis, Vol. 61, No. 6 | 15 November 2023
- Stability and discretization error analysis for the Cahn–Hilliard system via relative energy estimatesESAIM: Mathematical Modelling and Numerical Analysis, Vol. 57, No. 3 | 12 May 2023
- Fully Discrete Discontinuous Galerkin Numerical Scheme with Second-Order Temporal Accuracy for the Hydrodynamically Coupled Lipid Vesicle ModelJournal of Scientific Computing, Vol. 95, No. 1 | 15 February 2023
- Numerical analysis of energy stable weak Galerkin schemes for the Cahn–Hilliard equationCommunications in Nonlinear Science and Numerical Simulation, Vol. 118 | 1 Apr 2023
- An upwind DG scheme preserving the maximum principle for the convective Cahn-Hilliard modelNumerical Algorithms, Vol. 92, No. 3 | 12 August 2022
- A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equationsNumerical Methods for Partial Differential Equations, Vol. 39, No. 2 | 15 November 2022
- On the Superconvergence of a Hybridizable Discontinuous Galerkin Method for the Cahn–Hilliard EquationSIAM Journal on Numerical Analysis, Vol. 61, No. 1 | 3 February 2023
- A fully-decoupled discontinuous Galerkin approximation of the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth modelESAIM: Mathematical Modelling and Numerical Analysis, Vol. 56, No. 6 | 1 December 2022
- Comparison of an h - and hp -adaptive finite element solver for chemo-mechanically coupled battery electrode particlesExamples and Counterexamples, Vol. 2 | 1 Nov 2022
- The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A ReviewThe Virtual Element Method and its Applications | 8 May 2022
- $ C^1 $-VEM for some variants of the Cahn-Hilliard equation: A numerical explorationDiscrete and Continuous Dynamical Systems - S, Vol. 15, No. 8 | 1 Jan 2022
- Numerical Analysis of a Model of Two Phase Compressible Fluid FlowJournal of Scientific Computing, Vol. 89, No. 1 | 25 August 2021
- A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equationJournal of Computational Physics, Vol. 442 | 1 Oct 2021
- Linear energy stable and maximum principle preserving semi-implicit scheme for Allen–Cahn equation with double well potentialCommunications in Nonlinear Science and Numerical Simulation, Vol. 98 | 1 Jul 2021
- Modeling Refugee Movement Based on a Continuum Mechanics Phase-Field Approach of Porous MediaSIAM Journal on Applied Mathematics, Vol. 81, No. 5 | 20 September 2021
- A high order spectral difference-based phase field lattice Boltzmann method for incompressible two-phase flowsPhysics of Fluids, Vol. 32, No. 12 | 11 December 2020
- Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equationNumerical Algorithms, Vol. 85, No. 2 | 12 November 2019
- Numerical simulation of jet mode in electrospraying of Newtonian and viscoelastic fluidsInternational Journal of Multiphase Flow, Vol. 129 | 1 Aug 2020
- Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flowsNumerische Mathematik, Vol. 145, No. 1 | 26 March 2020
- Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins PotentialJournal of Scientific Computing, Vol. 82, No. 2 | 18 January 2020
- Bound-preserving flux limiting schemes for DG discretizations of conservation laws with applications to the Cahn–Hilliard equationComputer Methods in Applied Mechanics and Engineering, Vol. 359 | 1 Feb 2020
- The phase field method for geometric moving interfaces and their numerical approximationsGeometric Partial Differential Equations - Part I | 1 Jan 2020
- A fully discrete virtual element scheme for the Cahn–Hilliard equation in mixed formComputer Physics Communications, Vol. 246 | 1 Jan 2020
- Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equationNumerical Methods for Partial Differential Equations, Vol. 35, No. 4 | 23 February 2019
- Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn‐Hilliard equationsInternational Journal for Numerical Methods in Fluids, Vol. 90, No. 6 | 4 March 2019
- Discontinuous Galerkin finite element discretization of a degenerate Cahn–Hilliard equation with a single-well potentialCalcolo, Vol. 56, No. 2 | 4 April 2019
- A virtual element method for the Cahn–Hilliard problem in mixed formApplied Mathematics Letters, Vol. 87 | 1 Jan 2019
- Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equationESAIM: Mathematical Modelling and Numerical Analysis, Vol. 52, No. 3 | 13 September 2018
- A finite volume / discontinuous Galerkin method for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imagingComputational Geosciences, Vol. 22, No. 2 | 16 January 2018
- Weighted interior penalty discretization of fully nonlinear and weakly dispersive free surface shallow water flowsJournal of Computational Physics, Vol. 355 | 1 Feb 2018
- A Cahn‐Hilliard–type equation with application to tumor growth dynamicsMathematical Methods in the Applied Sciences, Vol. 40, No. 18 | 24 August 2017
- Inexact hierarchical scale separation: A two-scale approach for linear systems from discontinuous Galerkin discretizationsComputers & Mathematics with Applications, Vol. 74, No. 8 | 1 Oct 2017
- Stable finite element approximation of a Cahn–Hilliard–Stokes system coupled to an electric fieldEuropean Journal of Applied Mathematics, Vol. 28, No. 3 | 9 September 2016
- Error estimates of fully discrete finite element solutions for the 2D Cahn–Hilliard equation with infinite time horizonNumerical Methods for Partial Differential Equations, Vol. 33, No. 3 | 17 November 2016
- Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw systemNumerische Mathematik, Vol. 135, No. 3 | 2 June 2016
- A Discontinuous Galerkin Finite Element Framework for the Direct Numerical Simulation of Flow on High-Resolution Pore-Scale ImagesDay 1 Mon, February 20, 2017 | 20 February 2017
- The Cahn–Hilliard equation and some of its variantsAIMS Mathematics, Vol. 2, No. 3 | 1 Jan 2017
- Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equationIMA Journal of Numerical Analysis, Vol. 36, No. 4 | 17 December 2015
- Discontinuous Galerkin Approximation of Linear Parabolic Problems with Dynamic Boundary ConditionsJournal of Scientific Computing, Vol. 66, No. 3 | 25 June 2015
- A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal MeshesSIAM Journal on Numerical Analysis, Vol. 54, No. 1 | 6 January 2016
- A Hybrid High-Order Method for the Cahn--Hilliard problem in Mixed FormSIAM Journal on Numerical Analysis, Vol. 54, No. 3 | 21 June 2016
- Isogeometric analysis of the Cahn–Hilliard equation – a convergence studyJournal of Computational Physics, Vol. 305 | 1 Jan 2016
- Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based ModelsArchives of Computational Methods in Engineering, Vol. 22, No. 2 | 17 May 2014
- Second order schemes and time-step adaptivity for Allen–Cahn and Cahn–Hilliard modelsComputers & Mathematics with Applications, Vol. 68, No. 8 | 1 Oct 2014
- A fully discrete C0 interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equationJournal of Computational and Applied Mathematics, Vol. 247 | 1 Aug 2013
- On linear schemes for a Cahn–Hilliard diffuse interface modelJournal of Computational Physics, Vol. 234 | 1 Feb 2013
- A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solverDiscrete & Continuous Dynamical Systems - B, Vol. 18, No. 9 | 1 Jan 2013
- A three level linearized compact difference scheme for the Cahn-Hilliard equationScience China Mathematics, Vol. 55, No. 4 | 31 August 2011
- A Quadratic $C^0$ Interior Penalty Method for Linear Fourth Order Boundary Value Problems with Boundary Conditions of the Cahn--Hilliard TypeSIAM Journal on Numerical Analysis, Vol. 50, No. 4 | 21 August 2012
- The Cahn-Hilliard Equation with Logarithmic PotentialsMilan Journal of Mathematics, Vol. 79, No. 2 | 28 August 2011
- A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximationZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 91, No. 8 | 1 February 2011
- Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transitionNumerical Methods for Partial Differential Equations, Vol. 27, No. 1 | 27 October 2010

## View Options

### Get Access

**Access via your Institution**- Questions about how to access this content? Contact SIAM at
**[email protected]**.