Close Search Advanced
Journals
Resources
About Us
Open Access

A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations

A priori Error Analysis of a Discontinuous Galerkin Method for Cahn–Hilliard–Navier–Stokes Equations
Preview
Add to basket

Year:    2020

Author:    Chen Liu, Béatrice Rivière

CSIAM Transactions on Applied Mathematics, Vol. 1 (2020), Iss. 1 : pp. 104–141

Abstract

In this paper, we analyze an interior penalty discontinuous Galerkin method for solving the coupled Cahn–Hilliard and Navier–Stokes equations. We prove unconditional unique solvability of the discrete system, and we derive stability bounds without any restrictions on the chemical energy density function. The numerical solutions satisfy a discrete energy dissipation law and mass conservation laws. Convergence of the method is obtained by obtaining optimal a priori error estimates.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/csiam-am.2020-0005

CSIAM Transactions on Applied Mathematics, Vol. 1 (2020), Iss. 1 : pp. 104–141

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    38

Keywords:    Cahn–Hilliard–Navier–Stokes interior penalty discontinuous Galerkin method existence uniqueness stability error estimates.

Author Details

Chen Liu

Béatrice Rivière

  1. Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes

    Li, Rui | Gao, Yali | Chen, Zhangxin

    Numerical Algorithms, Vol. 95 (2024), Iss. 4 P.1981

    https://doi.org/10.1007/s11075-023-01635-5 [Citations: 1]

  2. A novel discontinuous Galerkin projection scheme for the hydrodynamics of nematic liquid crystals

    Zheng, Zhihui | Zou, Guang-an | Wang, Bo

    Communications in Nonlinear Science and Numerical Simulation, Vol. 137 (2024), Iss. P.108163

    https://doi.org/10.1016/j.cnsns.2024.108163 [Citations: 0]

  3. A Crank-Nicolson discontinuous Galerkin pressure-projection method for the hydrodynamic and sediment transport model

    Ren, Jinmiao | Wang, Bo | Zou, Guang-an

    Computers & Mathematics with Applications, Vol. 142 (2023), Iss. P.175

    https://doi.org/10.1016/j.camwa.2023.04.031 [Citations: 1]

  4. A splitting discontinuous Galerkin projection method for the magneto-hydrodynamic equations

    Wei, Yuanhong | Zou, Guang-an

    Applied Numerical Mathematics, Vol. 197 (2024), Iss. P.363

    https://doi.org/10.1016/j.apnum.2023.12.003 [Citations: 4]

  5. Unconditionally energy-stable discontinuous Galerkin method for the chemo-repulsion-Navier-Stokes system

    Wang, Meiting | Zou, Guang-an | Wang, Bo | Zhao, Wenju

    Computers & Mathematics with Applications, Vol. 150 (2023), Iss. P.132

    https://doi.org/10.1016/j.camwa.2023.09.012 [Citations: 1]

  6. A fully-decoupled discontinuous Galerkin method for the nematic liquid crystal flows with SAV approach

    Zheng, Zhihui | Zou, Guang-an | Wang, Bo | Zhao, Wenju

    Journal of Computational and Applied Mathematics, Vol. 429 (2023), Iss. P.115207

    https://doi.org/10.1016/j.cam.2023.115207 [Citations: 9]

  7. A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn–Hilliard–Navier–Stokes system

    Brunk, A. | Egger, H. | Habrich, O. | Lukáčová-Medviďová, M.

    Mathematical Models and Methods in Applied Sciences, Vol. 33 (2023), Iss. 12 P.2587

    https://doi.org/10.1142/S0218202523500562 [Citations: 1]

  8. Fully Discrete Discontinuous Galerkin Numerical Scheme with Second-Order Temporal Accuracy for the Hydrodynamically Coupled Lipid Vesicle Model

    Zou, Guang-an | Li, Zhaohua | Yang, Xiaofeng

    Journal of Scientific Computing, Vol. 95 (2023), Iss. 1

    https://doi.org/10.1007/s10915-023-02129-1 [Citations: 15]

  9. A stabilized Gauge-Uzawa discontinuous Galerkin method for the magneto-hydrodynamic equations

    Zou, Guang-an | Wei, Yuanhong | Yang, Xiaofeng

    BIT Numerical Mathematics, Vol. 64 (2024), Iss. 4

    https://doi.org/10.1007/s10543-024-01044-7 [Citations: 0]

  10. Efficient interior penalty discontinuous Galerkin projection method with unconditional energy stability and second-order temporal accuracy for the incompressible magneto-hydrodynamic system

    Zou, Guang-an | Wang, Bo | Yang, Xiaofeng

    Journal of Computational Physics, Vol. 495 (2023), Iss. P.112562

    https://doi.org/10.1016/j.jcp.2023.112562 [Citations: 8]

  11. Convergence of a Decoupled Splitting Scheme for the Cahn–Hilliard–Navier–Stokes System

    Liu, Chen | Masri, Rami | Riviere, Beatrice

    SIAM Journal on Numerical Analysis, Vol. 61 (2023), Iss. 6 P.2651

    https://doi.org/10.1137/22M1528069 [Citations: 5]

  12. A pressure-correction and bound-preserving discretization of the phase-field method for variable density two-phase flows

    Liu, Chen | Ray, Deep | Thiele, Christopher | Lin, Lu | Riviere, Beatrice

    Journal of Computational Physics, Vol. 449 (2022), Iss. P.110769

    https://doi.org/10.1016/j.jcp.2021.110769 [Citations: 7]

  13. Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem

    Kirk, Keegan L A | Riviere, Beatrice | Masri, Rami

    IMA Journal of Numerical Analysis, Vol. 44 (2024), Iss. 5 P.2752

    https://doi.org/10.1093/imanum/drad075 [Citations: 0]

  14. A rotational pressure-correction discontinuous Galerkin scheme for the Cahn-Hilliard-Darcy-Stokes system

    Wang, Meiting | Zou, Guang-an | Li, Jian

    Advances in Computational Mathematics, Vol. 50 (2024), Iss. 3

    https://doi.org/10.1007/s10444-024-10151-6 [Citations: 1]