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arxiv: 2604.21719 · v1 · submitted 2026-04-23 · 🧮 math.NA · cs.NA

A superconvergent hybridizable discontinuous Galerkin method for the convective Cahn--Hilliard equation

Gang Chen , Daozhi Han , Jiaxuan Liu , Yangwen Zhang , Dujin Zuo This is my paper

Pith reviewed 2026-05-09 21:06 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords hybridizable discontinuous GalerkinCahn-Hilliard equationconvex-concave splittingsuperconvergenceconvective flowsunconditional stability
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The pith

An HDG method for the convective Cahn-Hilliard equation achieves unconditional stability and optimal L2 convergence for scalar and flux variables at any polynomial degree by treating convection explicitly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a hybridizable discontinuous Galerkin discretization that pairs convex-concave splitting for the nonlinear terms with explicit treatment of convection. This combination removes the need for stabilization while keeping the scheme unconditionally stable and producing a symmetric reduced system after local elimination. Optimal L2 error bounds are proved for both the solution and the flux for every polynomial degree k at least zero by means of a specially constructed elliptic projection. The scalar unknowns further exhibit superconvergence once the local variables are eliminated. Numerical tests confirm the rates and illustrate practical performance on interface evolution problems.

Core claim

The proposed HDG scheme, combined with convex-concave splitting, discretizes the convection term explicitly without stabilization and establishes optimal L2-norm convergence rates for both scalar and flux variables for any k greater than or equal to zero through a specialized HDG elliptic projection; after local elimination the scalar variables superconverge.

What carries the argument

The specialized HDG elliptic projection operator, which supplies the approximation properties required to recover optimal L2 estimates while preserving the energy-dissipation structure of the convex-concave splitting.

If this is right

  • The scheme remains stable for arbitrary time-step sizes independent of the convection strength.
  • Piecewise-constant approximations retain their optimal convergence rate.
  • Local elimination produces a symmetric positive-definite system solvable by minimal-residual methods.
  • Scalar unknowns converge at a higher rate than the standard optimal order after static condensation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same explicit-convection treatment could be examined on convection-dominated phase-field models to test whether stability persists without added numerical diffusion.
  • Superconvergence of the scalar variable may permit coarser meshes when only interface location is required.

Load-bearing premise

The exact solution must be smooth enough and the domain regular enough for the projection operator to deliver the stated approximation properties.

What would settle it

Numerical experiments on a domain with corners or with a solution of limited regularity that produce convergence rates strictly below the predicted optimal order would show the analysis does not hold.

Figures

Figures reproduced from arXiv: 2604.21719 by Daozhi Han, Dujin Zuo, Gang Chen, Jiaxuan Liu, Yangwen Zhang.

Figure 1
Figure 1. Figure 1: Snapshots of the evolution of an initially cross-shaped profile under circular convection [PITH_FULL_IMAGE:figures/full_fig_p030_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of an initially cross-shaped profile under circular convection, computed with [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Snapshots of bulk regions under circular convection at selected times, computed with the [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of bulk regions under circular convection, computed with the upwind operator [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
read the original abstract

We propose a hybridizable discontinuous Galerkin (HDG) method combined with convex-concave splitting for the temporal discretization of the convective Cahn-Hilliard equation. The convection term is discretized explicitly without stabilization, yielding three key advantages: (1) unconditional stability, (2) preservation of the optimal convergence rate for piecewise constant approximations, and (3) a symmetric system after local elimination, enabling efficient solver via minimal residual methods. We establish optimal convergence rates in the $L^2$ norm for both the scalar and flux variables for any polynomial degree $k \geq 0$. To achieve optimal $L^2$-norm estimates, we introduce a specialized HDG elliptic projection operator and analyze its approximation properties. Within the HDG framework, local elimination is employed to reduce the degrees of freedom associated with the globally coupled unknowns, and the scalar variables exhibit superconvergence. Finally, numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a hybridizable discontinuous Galerkin (HDG) method for the convective Cahn-Hilliard equation, using convex-concave splitting for time discretization and explicit treatment of the convection term without stabilization. Key claims include unconditional stability, optimal L^2-norm convergence rates for both scalar and flux variables for any polynomial degree k ≥ 0, and superconvergence of the scalar variables. A specialized HDG elliptic projection operator is introduced to obtain the optimal estimates, local elimination reduces globally coupled unknowns, and numerical experiments are used to validate the theory.

Significance. If the unconditional stability and optimal/superconvergent rates hold, the method would offer a practical, efficient, and high-order accurate scheme for phase-field models with convection, which arise in materials science and multiphase flows. The symmetric linear systems after elimination and the preservation of rates for k=0 are notable practical strengths. The combination of HDG projection techniques with energy-dissipative splitting provides a template that could extend to related nonlinear parabolic problems.

major comments (3)
  1. [Error analysis (projection and convergence sections)] The optimal L^2 convergence claims for scalar and flux variables (abstract) rest on the approximation properties of the specialized HDG elliptic projection. These bounds implicitly require the exact solution to lie in H^{k+2} or higher (as is standard for such projections to recover full order), yet typical Cahn-Hilliard solutions with diffuse interfaces possess only limited regularity. This assumption is load-bearing for both the optimal-rate and superconvergence statements; the manuscript should either weaken the regularity hypothesis or provide a separate analysis under reduced regularity.
  2. [Stability and error analysis sections] The explicit, unstabilized discretization of the convection term is asserted to preserve unconditional stability and optimal rates even for k=0. However, the absorption of the convective contribution into the error estimates (via the energy-dissipation identity from the splitting) requires explicit control that does not degrade the rates for large convection coefficients or under mesh refinement. The current argument appears to rely on the same high-regularity projection; a concrete bound or counter-example test is needed.
  3. [Abstract and superconvergence discussion] Superconvergence of the scalar variable is claimed after local elimination, but the precise rate (e.g., one order higher than optimal) and the post-processing or projection step that delivers it are not quantified in the abstract or summary of results. Because superconvergence is a central advertised feature, the manuscript must state the exact superconvergent order and confirm it is not an artifact of the projection choice.
minor comments (2)
  1. The abstract states that 'the scalar variables exhibit superconvergence' without specifying the achieved order; adding the precise rate would improve clarity for readers.
  2. Notation for the specialized HDG elliptic projection operator should be introduced with a brief definition or reference to its defining equations at first use, rather than only in the analysis section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and valuable suggestions. We address each major comment below and indicate the revisions we will make to improve the manuscript.

read point-by-point responses

  1. Referee: The optimal L^2 convergence claims for scalar and flux variables (abstract) rest on the approximation properties of the specialized HDG elliptic projection. These bounds implicitly require the exact solution to lie in H^{k+2} or higher (as is standard for such projections to recover full order), yet typical Cahn-Hilliard solutions with diffuse interfaces possess only limited regularity. This assumption is load-bearing for both the optimal-rate and superconvergence statements; the manuscript should either weaken the regularity hypothesis or provide a separate analysis under reduced regularity.

    Authors: We acknowledge that our error analysis in Sections 4 and 5 relies on the assumption that the exact solution has sufficient regularity (u ∈ H^{k+2}(Ω) and similar for other variables) to obtain the optimal approximation properties of the HDG elliptic projection. This is a standard assumption in the analysis of high-order DG methods to achieve full order. For the convective Cahn-Hilliard equation, while solutions may exhibit limited regularity near interfaces, the numerical experiments in Section 6 confirm that the observed convergence rates match the theoretical predictions even for problems with sharp transitions. We will add a clarifying remark in the introduction and at the beginning of the error analysis section to explicitly state the regularity hypothesis and note that the method performs well numerically under reduced regularity. A complete analysis for lower regularity cases is left for future investigation. revision: partial

  2. Referee: The explicit, unstabilized discretization of the convection term is asserted to preserve unconditional stability and optimal rates even for k=0. However, the absorption of the convective contribution into the error estimates (via the energy-dissipation identity from the splitting) requires explicit control that does not degrade the rates for large convection coefficients or under mesh refinement. The current argument appears to rely on the same high-regularity projection; a concrete bound or counter-example test is needed.

    Authors: The unconditional stability result (Theorem 3.1) is derived from the convex-concave splitting and the discrete energy dissipation law, where the explicit convection term is incorporated via integration by parts without introducing instability, as the velocity is assumed divergence-free or bounded. In the error analysis (Theorem 5.2), the convective terms are estimated using the projection error and bounded by the energy norm, without rate degradation under the given assumptions. To provide concrete evidence, we will add numerical tests in the revised manuscript with varying convection strengths (large Peclet numbers) and successive mesh refinements, demonstrating that the convergence rates remain optimal for k=0 and higher. This will supplement the theoretical argument. revision: partial

  3. Referee: Superconvergence of the scalar variable is claimed after local elimination, but the precise rate (e.g., one order higher than optimal) and the post-processing or projection step that delivers it are not quantified in the abstract or summary of results. Because superconvergence is a central advertised feature, the manuscript must state the exact superconvergent order and confirm it is not an artifact of the projection choice.

    Authors: The superconvergence of the scalar variable is established in Section 5.3: after solving the local problems and eliminating the flux and auxiliary variables, the globally coupled scalar variable converges at rate O(h^{k+2}) in the L^2 norm, which is one order higher than the optimal rate O(h^{k+1}). This follows from the approximation properties of the specialized HDG projection, where the consistency error and the projection error combine to yield the higher order for the scalar. It is not an artifact, as the analysis includes all consistency terms from the HDG formulation. We will revise the abstract to include: 'the scalar variable exhibits superconvergence of order k+2 in the L^2-norm.' This quantifies the claim as requested. revision: yes

Circularity Check

0 steps flagged

No circularity: standard projection-based error analysis for HDG method

full rationale

The derivation proceeds by defining an HDG discretization with explicit convection treatment, introducing a specialized elliptic projection operator whose approximation properties are established separately under stated regularity assumptions, and then combining those bounds with the energy-dissipation identity from convex-concave splitting to obtain L2 error estimates and superconvergence via local elimination. None of these steps reduce by construction to the target convergence rates; the projection error bounds are derived from standard approximation theory rather than fitted to the method's output, and no load-bearing self-citations or ansatzes are invoked to force the result. The analysis is therefore self-contained and independent of the claimed rates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard finite-element assumptions about solution regularity and the stability properties of the convex-concave splitting; no free parameters or new physical entities are introduced in the abstract.

axioms (2)
  • domain assumption The solution possesses sufficient regularity for the error estimates and projection approximation properties to hold.
    Required for all convergence analysis in Galerkin methods for nonlinear PDEs.
  • domain assumption The convex-concave splitting of the nonlinear term yields the energy dissipation needed for unconditional stability.
    Central to the stability claim when convection is treated explicitly.

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Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    Rafael Rodríguez-Galván

    Daniel Acosta-Soba, Francisco Guillén-González, and J. Rafael Rodríguez-Galván. An upwind DG scheme preserving the maximum principle for the convective Cahn-Hilliard model.Numer. Algorithms, 92(3):1589–1619, 2023

    work page 2023

  2. [2]

    Rafael Rodríguez-Galván, and Jin Wang

    Daniel Acosta-Soba, Francisco Guillén-González, J. Rafael Rodríguez-Galván, and Jin Wang. Property-preserving numerical approximation of a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility.Appl. Numer. Math., 209:68–83, 2025

    work page 2025

  3. [3]

    A. R. Appadu, J. K. Djoko, H. H. Gidey, and J. M. S. Lubuma. Analysis of multilevel finite volume approximation of 2D convective Cahn-Hilliard equation.Jpn. J. Ind. Appl. Math., 34(1):253–304, 2017

    work page 2017

  4. [4]

    Aristotelous, Ohannes A

    Andreas C. Aristotelous, Ohannes A. Karakashian, and Steven M. Wise. Adaptive, second- order in time, primitive-variable discontinuous Galerkin schemes for a Cahn-Hilliard equation with a mass source.IMA J. Numer. Anal., 35(3):1167–1198, 2015

    work page 2015

  5. [5]

    V. E. Badalassi, H. D. Ceniceros, and S. Banerjee. Computation of multiphase systems with phase field models.J. Comput. Phys., 190(2):371–397, 2003

    work page 2003

  6. [6]

    Barrett, James F

    John W. Barrett, James F. Blowey, and Harald Garcke. Finite element approximation of the Cahn-Hilliard equation with degenerate mobility.SIAM J. Numer. Anal., 37(1):286–318, 1999

    work page 1999

  7. [7]

    An implicit midpoint spectral approxi- mation of nonlocal Cahn-Hilliard equations.SIAM J

    Barbora Benešová, Christof Melcher, and Endre Süli. An implicit midpoint spectral approxi- mation of nonlocal Cahn-Hilliard equations.SIAM J. Numer. Anal., 52(3):1466–1496, 2014

    work page 2014

  8. [8]

    A numerical implementation of the finite- difference algorithm for solving conserved cahn–hilliard equation.Journal of Physics: Confer- ence Series, 1936(1):12014, 2021

    Wilcox Boma, Qinguy Wang, and Ayodeji Abiodun. A numerical implementation of the finite- difference algorithm for solving conserved cahn–hilliard equation.Journal of Physics: Confer- ence Series, 1936(1):12014, 2021

    work page 1936

  9. [9]

    Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model.Eur

    Franck Boyer, Laurent Chupin, and Pierre Fabrie. Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model.Eur. J. Mech. B Fluids, 23(5):759–780, 2004

    work page 2004

  10. [10]

    Brenner, Amanda E

    Susanne C. Brenner, Amanda E. Diegel, and Li-Yeng Sung. A robust solver for a mixed finite element method for the Cahn-Hilliard equation.J. Sci. Comput., 77(2):1234–1249, 2018

    work page 2018

  11. [11]

    Brenner, Amanda E

    Susanne C. Brenner, Amanda E. Diegel, and Li-Yeng Sung. A robust solver for a second order mixed finite element method for the Cahn-Hilliard equation.J. Comput. Appl. Math., 364:112322, 12, 2020. 39 G. Chen, D. Han, J. Liu, Y. Zhang and D. Zuo

    work page 2020

  12. [12]

    Brenner and L

    Susanne C. Brenner and L. Ridgway Scott.The mathematical theory of finite element methods, volume 15 ofTexts in Applied Mathematics. Springer, New York, third edition, 2008

    work page 2008

  13. [13]

    OptimalL2 error estimates of uncon- ditionally stable finite element schemes for the Cahn-Hilliard-Navier-Stokes system.SIAM J

    Wentao Cai, Weiwei Sun, Jilu Wang, and Zongze Yang. OptimalL2 error estimates of uncon- ditionally stable finite element schemes for the Cahn-Hilliard-Navier-Stokes system.SIAM J. Numer. Anal., 61(3):1218–1245, 2023

    work page 2023

  14. [14]

    Error estimates for a fully discretized scheme to a Cahn-Hilliard phase-field model for two-phase incompressible flows.Math

    Yongyong Cai and Jie Shen. Error estimates for a fully discretized scheme to a Cahn-Hilliard phase-field model for two-phase incompressible flows.Math. Comp., 87(313):2057–2090, 2018

    work page 2057

  15. [15]

    Analysis of HDG methods for Oseen equations.J

    Aycil Cesmelioglu, Bernardo Cockburn, Ngoc Cuong Nguyen, and Jaume Peraire. Analysis of HDG methods for Oseen equations.J. Sci. Comput., 55(2):392–431, 2013

    work page 2013

  16. [16]

    Di Pietro, and Fabien Marche

    Florent Chave, Daniele A. Di Pietro, and Fabien Marche. A hybrid high-order method for the convective Cahn-Hilliard problem in mixed form. InFinite volumes for complex applications VIII—hyperbolic, elliptic and parabolic problems, volume 200 ofSpringer Proc. Math. Stat., pages 517–525. Springer, Cham, 2017

    work page 2017

  17. [17]

    On the superconvergence of a hydridizable discontinuous galerkin method for the cahn-hilliard equation.arXiv:1901.00079, 2019

    Gang Chen, Daozhi Han, John Singler, and Yangwen Zhang. On the superconvergence of a hydridizable discontinuous galerkin method for the cahn-hilliard equation.arXiv:1901.00079, 2019

    work page arXiv 1901

  18. [18]

    Singler, and Yangwen Zhang

    Gang Chen, Daozhi Han, John R. Singler, and Yangwen Zhang. On the superconvergence of a hybridizable discontinuous Galerkin method for the Cahn-Hilliard equation.SIAM J. Numer. Anal., 61(1):83–109, 2023

    work page 2023

  19. [19]

    Singler, Yangwen Zhang, and Xiaobo Zheng

    Gang Chen, Weiwei Hu, Jiguang Shen, John R. Singler, Yangwen Zhang, and Xiaobo Zheng. An HDG method for distributed control of convection diffusion PDEs.J. Comput. Appl. Math., 343:643–661, 2018

    work page 2018

  20. [20]

    Singler, and Yangwen Zhang

    Gang Chen, John R. Singler, and Yangwen Zhang. An HDG method for Dirichlet boundary control of convection dominated diffusion PDEs.SIAM J. Numer. Anal., 57(4):1919–1946, 2019

    work page 1919

  21. [21]

    Robusta posteriorierror estimates for HDG method for convection-diffusion equations.IMA J

    Huangxin Chen, Jingzhi Li, and Weifeng Qiu. Robusta posteriorierror estimates for HDG method for convection-diffusion equations.IMA J. Numer. Anal., 36(1):437–462, 2016

    work page 2016

  22. [22]

    Analysis of variable-degree HDG methods for convection- diffusion equations

    Yanlai Chen and Bernardo Cockburn. Analysis of variable-degree HDG methods for convection- diffusion equations. Part I: general nonconforming meshes.IMA J. Numer. Anal., 32(4):1267– 1293, 2012

    work page 2012

  23. [23]

    Analysis of variable-degree HDG methods for convection- diffusion equations

    Yanlai Chen and Bernardo Cockburn. Analysis of variable-degree HDG methods for convection- diffusion equations. Part II: Semimatching nonconforming meshes.Math. Comp., 83(285):87– 111, 2014

    work page 2014

  24. [24]

    Kelong Cheng, Wenqiang Feng, Cheng Wang, and Steven M. Wise. An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation.J. Comput. Appl. Math., 362:574– 595, 2019

    work page 2019

  25. [25]

    Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems.SIAM J

    Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems.SIAM J. Numer. Anal., 47(2):1319–1365, 2009. 40 An HDG method for convective Cahn–Hilliard equations

    work page 2009

  26. [26]

    Analysis of HDG methods for Stokes flow.Math

    Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, and Francisco-Javier Sayas. Analysis of HDG methods for Stokes flow.Math. Comp., 80(274):723– 760, 2011

    work page 2011

  27. [27]

    A time-adaptive finite volume method for the Cahn- Hilliard and Kuramoto-Sivashinsky equations.J

    Luis Cueto-Felgueroso and Jaume Peraire. A time-adaptive finite volume method for the Cahn- Hilliard and Kuramoto-Sivashinsky equations.J. Comput. Phys., 227(24):9985–10017, 2008

    work page 2008

  28. [28]

    Dargaville and T

    S. Dargaville and T. W. Farrell. A least squares based finite volume method for the Cahn- Hilliard and Cahn-Hilliard-reaction equations.J. Comput. Appl. Math., 273:225–244, 2015

    work page 2015

  29. [29]

    Di Pietro and Jérôme Droniou

    Daniele A. Di Pietro and Jérôme Droniou. A third Strang lemma and an Aubin-Nitsche trick for schemes in fully discrete formulation.Calcolo, 55(3):Paper No. 40, 39, 2018

    work page 2018

  30. [30]

    Modeling, Simulation and Applications

    Daniele Antonio Di Pietro and Jérôme Droniou.The hybrid high-order method for polytopal meshes, volume 19 ofMS&A. Modeling, Simulation and Applications. Springer, Cham, 2020

    work page 2020

  31. [31]

    Diegel, Cheng Wang, and Steven M

    Amanda E. Diegel, Cheng Wang, and Steven M. Wise. Stability and convergence of a second- order mixed finite element method for the Cahn-Hilliard equation.IMA J. Numer. Anal., 36(4):1867–1897, 2016

    work page 2016

  32. [32]

    Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems.SIAM J

    Bo Dong and Chi-Wang Shu. Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems.SIAM J. Numer. Anal., 47(5):3240–3268, 2009

    work page 2009

  33. [33]

    Elliott and Donald A

    Charles M. Elliott and Donald A. French. A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation.SIAM J. Numer. Anal., 26(4):884–903, 1989

    work page 1989

  34. [34]

    Karakashian

    Xiaobing Feng and Ohannes A. Karakashian. Fully discrete dynamic mesh discontinu- ous Galerkin methods for the Cahn-Hilliard equation of phase transition.Math. Comp., 76(259):1093–1117, 2007

    work page 2007

  35. [35]

    A fully discrete mixed finite element method for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise.J

    Xiaobing Feng, Yukun Li, and Yi Zhang. A fully discrete mixed finite element method for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise.J. Sci. Comput., 83(1):Paper No. 23, 24, 2020

    work page 2020

  36. [36]

    Phase separation in the advective Cahn-Hilliard equation.J

    Yu Feng, Yuanyuan Feng, Gautam Iyer, and Jean-Luc Thiffeault. Phase separation in the advective Cahn-Hilliard equation.J. Nonlinear Sci., 30(6):2821–2845, 2020

    work page 2020

  37. [37]

    Alpak, and Beatrice Riviere

    Florian Frank, Chen Liu, Faruk O. Alpak, and Beatrice Riviere. A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging.Comput. Geosci., 22(2):543–563, 2018

    work page 2018

  38. [38]

    A divergence-free HDG scheme for the Cahn-Hilliard phase-field model for two- phase incompressible flow.J

    Guosheng Fu. A divergence-free HDG scheme for the Cahn-Hilliard phase-field model for two- phase incompressible flow.J. Comput. Phys., 419:109671, 16, 2020

    work page 2020

  39. [39]

    An analysis of HDG methods for convection- dominated diffusion problems.ESAIM Math

    Guosheng Fu, Weifeng Qiu, and Wujun Zhang. An analysis of HDG methods for convection- dominated diffusion problems.ESAIM Math. Model. Numer. Anal., 49(1):225–256, 2015

    work page 2015

  40. [40]

    Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

    Takeshi Fukao, Shuji Yoshikawa, and Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Commun. Pure Appl. Anal., 16(5):1915–1938, 2017

    work page 1915

  41. [41]

    H. H. Gidey and B. D. Reddy. Operator-splitting methods for the 2D convective Cahn-Hilliard equation.Comput. Math. Appl., 77(12):3128–3153, 2019. 41 G. Chen, D. Han, J. Liu, Y. Zhang and D. Zuo

    work page 2019

  42. [42]

    A new HDG method for Dirichlet boundary control of convection diffusion PDEs II: low regularity

    Wei Gong, Weiwei Hu, Mariano Mateos, John Singler, Xiao Zhang, and Yangwen Zhang. A new HDG method for Dirichlet boundary control of convection diffusion PDEs II: low regularity. SIAM J. Numer. Anal., 56(4):2262–2287, 2018

    work page 2018

  43. [43]

    Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations.J

    Jason Goulding, Mehrnaz Ayazi, Tamar Shinar, and Craig Schroeder. Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations.J. Comput. Phys., 519:Paper No. 113382, 35, 2024

    work page 2024

  44. [44]

    G. Grün. On convergent schemes for diffuse interface models for two-phase flow of incompress- ible fluids with general mass densities.SIAM J. Numer. Anal., 51(6):3036–3061, 2013

    work page 2013

  45. [45]

    An explicit adaptive finite difference method for the Cahn-Hilliard equation.J

    Seokjun Ham, Yibao Li, Darae Jeong, Chaeyoung Lee, Soobin Kwak, Youngjin Hwang, and Junseok Kim. An explicit adaptive finite difference method for the Cahn-Hilliard equation.J. Nonlinear Sci., 32(6):Paper No. 80, 19, 2022

    work page 2022

  46. [46]

    Density convergence of a fully discrete finite difference method for stochastic Cahn-Hilliard equation.Math

    Jialin Hong, Diancong Jin, and Derui Sheng. Density convergence of a fully discrete finite difference method for stochastic Cahn-Hilliard equation.Math. Comp., 93(349):2215–2264, 2024

    work page 2024

  47. [47]

    Singler, Yangwen Zhang, and Xiaobo Zheng

    Weiwei Hu, Jiguang Shen, John R. Singler, Yangwen Zhang, and Xiaobo Zheng. A supercon- vergent HDG method for distributed control of convection diffusion PDEs.J. Sci. Comput., 76(3):1436–1457, 2018

    work page 2018

  48. [48]

    PreconditionedSAV-leapfrogfinitedifferencemeth- ods for spatial fractional Cahn-Hilliard equations.Appl

    XinHuang, DongfangLi, andHai-WeiSun. PreconditionedSAV-leapfrogfinitedifferencemeth- ods for spatial fractional Cahn-Hilliard equations.Appl. Math. Lett., 138:Paper No. 108510, 7, 2023

    work page 2023

  49. [49]

    Calculation of two-phase Navier-Stokes flows using phase-field modeling.J

    David Jacqmin. Calculation of two-phase Navier-Stokes flows using phase-field modeling.J. Comput. Phys., 155(1):96–127, 1999

    work page 1999

  50. [50]

    Discontinuous Galerkin finite element approxima- tion of the Cahn-Hilliard equation with convection.SIAM J

    David Kay, Vanessa Styles, and Endre Süli. Discontinuous Galerkin finite element approxima- tion of the Cahn-Hilliard equation with convection.SIAM J. Numer. Anal., 47(4):2660–2685, 2009

    work page 2009

  51. [51]

    A diffuse-interface model for axisymmetric immiscible two-phase flow.Appl

    Junseok Kim. A diffuse-interface model for axisymmetric immiscible two-phase flow.Appl. Math. Comput., 160(2):589–606, 2005

    work page 2005

  52. [52]

    Conservative multigrid methods for Cahn-Hilliard fluids.J

    Junseok Kim, Kyungkeun Kang, and John Lowengrub. Conservative multigrid methods for Cahn-Hilliard fluids.J. Comput. Phys., 193(2):511–543, 2004

    work page 2004

  53. [53]

    Keegan L. A. Kirk, Beatrice Riviere, and Rami Masri. Numerical analysis of a hybridized dis- continuous Galerkin method for the Cahn-Hilliard problem.IMA J. Numer. Anal., 44(5):2752– 2792, 2024

    work page 2024

  54. [54]

    FastimplicitupdateschemesforCahn-Hilliard- type gradient flow in the context of Fourier-spectral methods.Comput

    A.Krischok, B.Yaraguntappa, andM.-A.Keip. FastimplicitupdateschemesforCahn-Hilliard- type gradient flow in the context of Fourier-spectral methods.Comput. Methods Appl. Mech. Engrg., 431:Paper No. 117220, 19, 2024

    work page 2024

  55. [55]

    On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations.J

    Dong Li and Zhonghua Qiao. On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations.J. Sci. Comput., 70(1):301–341, 2017

    work page 2017

  56. [56]

    Convergence of a decoupled splitting scheme for the Cahn-Hilliard-Navier-Stokes system.SIAM J

    Chen Liu, Rami Masri, and Beatrice Riviere. Convergence of a decoupled splitting scheme for the Cahn-Hilliard-Navier-Stokes system.SIAM J. Numer. Anal., 61(6):2651–2694, 2023. 42 An HDG method for convective Cahn–Hilliard equations

    work page 2023

  57. [57]

    A simple and efficient convex op- timization based bound-preserving high order accurate limiter for Cahn-Hilliard-Navier-Stokes system.SIAM J

    Chen Liu, Beatrice Riviere, Jie Shen, and Xiangxiong Zhang. A simple and efficient convex op- timization based bound-preserving high order accurate limiter for Cahn-Hilliard-Navier-Stokes system.SIAM J. Sci. Comput., 46(3):A1923–A1948, 2024

    work page 2024

  58. [58]

    Evans, Micheal J

    Ju Liu, Luca Dedè, John A. Evans, Micheal J. Borden, and Thomas J. R. Hughes. Isogeometric analysis of the advective Cahn-Hilliard equation: spinodal decomposition under shear flow.J. Comput. Phys., 242:321–350, 2013

    work page 2013

  59. [59]

    Medina, Elson M

    Emmanuel Y. Medina, Elson M. Toledo, Iury Igreja, and Bernardo M. Rocha. A stabilized hybrid discontinuous Galerkin method for the Cahn-Hilliard equation.J. Comput. Appl. Math., 406:Paper No. 114025, 16, 2022

    work page 2022

  60. [60]

    Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dy- namic boundary conditions.IMA J

    Flore Nabet. Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dy- namic boundary conditions.IMA J. Numer. Anal., 36(4):1898–1942, 2016

    work page 1942

  61. [61]

    An error estimate for a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions.Numer

    Flore Nabet. An error estimate for a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions.Numer. Math., 149(1):185–226, 2021

    work page 2021

  62. [62]

    An HDG method for convection diffusion equation.J

    Weifeng Qiu and Ke Shi. An HDG method for convection diffusion equation.J. Sci. Comput., 66(1):346–357, 2016

    work page 2016

  63. [63]

    A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes.IMA J

    Weifeng Qiu and Ke Shi. A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes.IMA J. Numer. Anal., 36(4):1943–1967, 2016

    work page 1943

  64. [64]

    A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains.J

    Sander Rhebergen and Bernardo Cockburn. A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains.J. Comput. Phys., 231(11):4185–4204, 2012

    work page 2012

  65. [65]

    Sander Rhebergen, Bernardo Cockburn, and Jaap J. W. van der Vegt. A space-time discon- tinuous Galerkin method for the incompressible Navier-Stokes equations.J. Comput. Phys., 233:339–358, 2013

    work page 2013

  66. [66]

    Postprocessing mixed finite element methods for solving Cahn-Hilliard equation: methods and error analysis.J

    Wansheng Wang, Long Chen, and Jie Zhou. Postprocessing mixed finite element methods for solving Cahn-Hilliard equation: methods and error analysis.J. Sci. Comput., 67(2):724–746, 2016

    work page 2016

  67. [67]

    Shilin Zeng, Ziqing Xie, Xiaofeng Yang, and Jiangxing Wang. Fully discrete, decoupled and energy-stable Fourier-spectral numerical scheme for the nonlocal Cahn-Hilliard equation cou- pledwithNavier-Stokes/Darcyflowregimeoftwo-phaseincompressibleflows.Comput. Methods Appl. Mech. Engrg., 415:Paper No. 116289, 23, 2023

    work page 2023

  68. [68]

    A nonconforming finite element method for the Cahn-Hilliard equation.J

    Shuo Zhang and Ming Wang. A nonconforming finite element method for the Cahn-Hilliard equation.J. Comput. Phys., 229(19):7361–7372, 2010

    work page 2010

  69. [69]

    Fourier spectral approximation to global attractor for 2D convective Cahn- Hilliard equation.Bull

    Xiaopeng Zhao. Fourier spectral approximation to global attractor for 2D convective Cahn- Hilliard equation.Bull. Malays. Math. Sci. Soc., 41(2):1119–1138, 2018

    work page 2018

  70. [70]

    Energy stability and convergence of the scalar auxiliary variable Fourier-spectral method for the viscous Cahn-Hilliard equation.Numer

    Nan Zheng and Xiaoli Li. Energy stability and convergence of the scalar auxiliary variable Fourier-spectral method for the viscous Cahn-Hilliard equation.Numer. Methods Partial Dif- ferential Equations, 36(5):998–1011, 2020. 43

    work page 2020