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doi: 10.3934/era.2021030

A C0 interior penalty method for the Cahn-Hilliard equation

Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA

* Corresponding author: Amanda E. Diegel

Received  November 2020 Revised  February 2021 Published  April 2021

We present a C$ ^0 $ interior penalty method for the Cahn-Hilliard equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C$ ^0 $ interior penalty finite element method developed for the Cahn-Hilliard equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C$ ^0 $ interior penalty method. We furthermore support our conclusions with a few numerical experiments.

Citation: Amanda E. Diegel. A C0 interior penalty method for the Cahn-Hilliard equation. Electronic Research Archive, doi: 10.3934/era.2021030
References:
[1]

P. F. AntoniettiL. B. Da VeigaS. Scacchi and M. Verani, A C^1 virtual element method for the Cahn–Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54 (2016), 34-56.  doi: 10.1137/15M1008117.  Google Scholar

[2]

A. C. AristotelousO. A. Karakashian and S. 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 (2015), 1167-1198.  doi: 10.1093/imanum/dru035.  Google Scholar

[3]

S. C. Brenner, $C^0$ interior penalty methods, in Frontiers in Numerical Analysis-Durham 2010, Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 85 (2012), 79–147. doi: 10.1007/978-3-642-23914-4_2.  Google Scholar

[4]

S. C. BrennerS. GuT. Gudi and L.-Y. Sung, A quadratic $C^0$ interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn-Hilliard type, SIAM J. Numer. Anal., 50 (2012), 2088-2110.  doi: 10.1137/110847469.  Google Scholar

[5]

S. C. BrennerT. Gudi and L.-Y. Sung, An a posteriori error estimator for a quadratic $C^0$-interior penalty method for the biharmonic problem, IMA J. Numer. Anal., 30 (2010), 777-798.  doi: 10.1093/imanum/drn057.  Google Scholar

[6]

S. C. Brenner and M. Neilan, A $C^0$ interior penalty method for a fourth order elliptic singular perturbation problem, SIAM J. Numer. Anal., 49 (2011), 869-892.  doi: 10.1137/100786988.  Google Scholar

[7]

S. C. Brenner and L.-Y. Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput., 22 (2005), 83-118.  doi: 10.1007/s10915-004-4135-7.  Google Scholar

[8]

S. C. Brenner and L.-Y. Sung, Multigrid algorithms for $C^0$ interior penalty methods, SIAM J. Numer. Anal., 44 (2006), 199-223.  doi: 10.1137/040611835.  Google Scholar

[9]

S. C. Brenner, L.-Y. Sung and Y. Zhang, A quadratic $C^0$ interior penalty method for an elliptic optimal control problem with state constraints, in Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, IMA Vol. Math. Appl., Springer, Cham, 157 (2014), 97–132. doi: 10.1007/978-3-319-01818-8_4.  Google Scholar

[10]

S. C. BrennerL.-Y. Sung and Y. Zhang, Post-processing procedures for an elliptic distributed optimal control problem with pointwise state constraints, Appl. Numer. Math., 95 (2015), 99-117.  doi: 10.1016/j.apnum.2015.03.001.  Google Scholar

[11]

S. C. BrennerL.-Y. SungH. Zhang and Y. Zhang, A quadratic $C^0$ interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates, SIAM J. Numer. Anal., 50 (2012), 3329-3350.  doi: 10.1137/110845926.  Google Scholar

[12]

S. C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for $C^0$ interior penalty methods, Numer. Math., 102 (2005), 231-255.  doi: 10.1007/s00211-005-0641-2.  Google Scholar

[13]

S. C. BrennerK. Wang and J. Zhao, Poincaré–Friedrichs inequalities for piecewise $H^2$ functions, Numer. Funct. Anal. Optim., 25 (2004), 463-478.  doi: 10.1081/NFA-200042165.  Google Scholar

[14]

J. W. Cahn, On spinodal decomposition, Acta Metall Mater, 9 (1961), 795-801.  doi: 10.1002/9781118788295.ch11.  Google Scholar

[15]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[16]

L. Chen, Direct solver for the Cahn–Hilliard equation by Legendre–Galerkin spectral method, J. Comput. Appl. Math., 358 (2019), 34-45.  doi: 10.1016/j.cam.2019.03.008.  Google Scholar

[17]

W. ChenX. WangY. Yan and Z. Zhang, A second order BDF numerical scheme with variable steps for the Cahn–Hilliard equation, SIAM J. Numer. Anal., 57 (2019), 495-525.  doi: 10.1137/18M1206084.  Google Scholar

[18]

K. ChengC. WangS. M. Wise and X. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn–Hilliard equation and its solution by the homogeneous linear iteration method, J. Sci. Comput., 69 (2016), 1083-1114.  doi: 10.1007/s10915-016-0228-3.  Google Scholar

[19]

A. E. DiegelX. H. Feng and S. M. Wise, Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system, SIAM J. Numer. Anal., 53 (2015), 127-152.  doi: 10.1137/130950628.  Google Scholar

[20]

A. E. DiegelC. Wang and S. M. Wise, Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation, IMA J. Numer. Anal., 36 (2016), 1867-1897.  doi: 10.1093/imanum/drv065.  Google Scholar

[21]

G. EngelK. GarikipatiT. J. R. HughesM. G. LarsonL. Mazzei and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3669-3750.  doi: 10.1016/S0045-7825(02)00286-4.  Google Scholar

[22]

T. FraunholzR. H. W. Hoppe and M. Peter, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the biharmonic problem, J. Numer. Math., 23 (2015), 317-330.  doi: 10.1515/jnma-2015-0021.  Google Scholar

[23]

T. Gudi and H. S. Gupta, A fully discrete $C^0$ interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation, J. Comput. Appl. Math., 247 (2013), 1-16.  doi: 10.1016/j.cam.2012.12.019.  Google Scholar

[24]

T. GudiH. S. Gupta and N. Nataraj, Analysis of an interior penalty method for fourth order problems on polygonal domains, J. Sci. Comput., 54 (2013), 177-199.  doi: 10.1007/s10915-012-9612-9.  Google Scholar

[25]

J. GuoC. WangS. M. Wise and X. Yue, An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation, Commun. Math. Sci., 14 (2016), 489-515.  doi: 10.4310/CMS.2016.v14.n2.a8.  Google Scholar

[26]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. Ⅰ. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.  doi: 10.1137/0719018.  Google Scholar

[27]

R. H. W. Hoppe and C. Linsenmann, $\rm C^0$-interior penalty discontinuous Galerkin approximation of a sixth-order Cahn-Hilliard equation, Contrib. PDEs Appl., 47 (2019), 297-325.   Google Scholar

[28]

S. Kaessmair and P. Steinmann, Comparative computational analysis of the Cahn–Hilliard equation with emphasis on $C^1$-continuous methods, J. Comput. Phys., 322 (2016), 783-803.  doi: 10.1016/j.jcp.2016.07.005.  Google Scholar

[29]

D. Li and Z. Qiao, On second order semi-implicit Fourier spectral methods for 2D Cahn–Hilliard equations, J. Sci. Comput., 70 (2017), 301-341.  doi: 10.1007/s10915-016-0251-4.  Google Scholar

[30]

A. Miranville, The Cahn–Hilliard Equation: Recent Advances and Applications, SIAM, Philadelphia, 2019. doi: 10.1137/1.9781611975925.  Google Scholar

[31]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2$^nd$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar

[32]

G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn–Hilliard equation and its applicability to related energy-based models, Arch. Comput. Methods Eng., 22 (2015), 269-289.  doi: 10.1007/s11831-014-9112-1.  Google Scholar

show all references

References:
[1]

P. F. AntoniettiL. B. Da VeigaS. Scacchi and M. Verani, A C^1 virtual element method for the Cahn–Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54 (2016), 34-56.  doi: 10.1137/15M1008117.  Google Scholar

[2]

A. C. AristotelousO. A. Karakashian and S. 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 (2015), 1167-1198.  doi: 10.1093/imanum/dru035.  Google Scholar

[3]

S. C. Brenner, $C^0$ interior penalty methods, in Frontiers in Numerical Analysis-Durham 2010, Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 85 (2012), 79–147. doi: 10.1007/978-3-642-23914-4_2.  Google Scholar

[4]

S. C. BrennerS. GuT. Gudi and L.-Y. Sung, A quadratic $C^0$ interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn-Hilliard type, SIAM J. Numer. Anal., 50 (2012), 2088-2110.  doi: 10.1137/110847469.  Google Scholar

[5]

S. C. BrennerT. Gudi and L.-Y. Sung, An a posteriori error estimator for a quadratic $C^0$-interior penalty method for the biharmonic problem, IMA J. Numer. Anal., 30 (2010), 777-798.  doi: 10.1093/imanum/drn057.  Google Scholar

[6]

S. C. Brenner and M. Neilan, A $C^0$ interior penalty method for a fourth order elliptic singular perturbation problem, SIAM J. Numer. Anal., 49 (2011), 869-892.  doi: 10.1137/100786988.  Google Scholar

[7]

S. C. Brenner and L.-Y. Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput., 22 (2005), 83-118.  doi: 10.1007/s10915-004-4135-7.  Google Scholar

[8]

S. C. Brenner and L.-Y. Sung, Multigrid algorithms for $C^0$ interior penalty methods, SIAM J. Numer. Anal., 44 (2006), 199-223.  doi: 10.1137/040611835.  Google Scholar

[9]

S. C. Brenner, L.-Y. Sung and Y. Zhang, A quadratic $C^0$ interior penalty method for an elliptic optimal control problem with state constraints, in Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, IMA Vol. Math. Appl., Springer, Cham, 157 (2014), 97–132. doi: 10.1007/978-3-319-01818-8_4.  Google Scholar

[10]

S. C. BrennerL.-Y. Sung and Y. Zhang, Post-processing procedures for an elliptic distributed optimal control problem with pointwise state constraints, Appl. Numer. Math., 95 (2015), 99-117.  doi: 10.1016/j.apnum.2015.03.001.  Google Scholar

[11]

S. C. BrennerL.-Y. SungH. Zhang and Y. Zhang, A quadratic $C^0$ interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates, SIAM J. Numer. Anal., 50 (2012), 3329-3350.  doi: 10.1137/110845926.  Google Scholar

[12]

S. C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for $C^0$ interior penalty methods, Numer. Math., 102 (2005), 231-255.  doi: 10.1007/s00211-005-0641-2.  Google Scholar

[13]

S. C. BrennerK. Wang and J. Zhao, Poincaré–Friedrichs inequalities for piecewise $H^2$ functions, Numer. Funct. Anal. Optim., 25 (2004), 463-478.  doi: 10.1081/NFA-200042165.  Google Scholar

[14]

J. W. Cahn, On spinodal decomposition, Acta Metall Mater, 9 (1961), 795-801.  doi: 10.1002/9781118788295.ch11.  Google Scholar

[15]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[16]

L. Chen, Direct solver for the Cahn–Hilliard equation by Legendre–Galerkin spectral method, J. Comput. Appl. Math., 358 (2019), 34-45.  doi: 10.1016/j.cam.2019.03.008.  Google Scholar

[17]

W. ChenX. WangY. Yan and Z. Zhang, A second order BDF numerical scheme with variable steps for the Cahn–Hilliard equation, SIAM J. Numer. Anal., 57 (2019), 495-525.  doi: 10.1137/18M1206084.  Google Scholar

[18]

K. ChengC. WangS. M. Wise and X. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn–Hilliard equation and its solution by the homogeneous linear iteration method, J. Sci. Comput., 69 (2016), 1083-1114.  doi: 10.1007/s10915-016-0228-3.  Google Scholar

[19]

A. E. DiegelX. H. Feng and S. M. Wise, Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system, SIAM J. Numer. Anal., 53 (2015), 127-152.  doi: 10.1137/130950628.  Google Scholar

[20]

A. E. DiegelC. Wang and S. M. Wise, Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation, IMA J. Numer. Anal., 36 (2016), 1867-1897.  doi: 10.1093/imanum/drv065.  Google Scholar

[21]

G. EngelK. GarikipatiT. J. R. HughesM. G. LarsonL. Mazzei and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3669-3750.  doi: 10.1016/S0045-7825(02)00286-4.  Google Scholar

[22]

T. FraunholzR. H. W. Hoppe and M. Peter, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the biharmonic problem, J. Numer. Math., 23 (2015), 317-330.  doi: 10.1515/jnma-2015-0021.  Google Scholar

[23]

T. Gudi and H. S. Gupta, A fully discrete $C^0$ interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation, J. Comput. Appl. Math., 247 (2013), 1-16.  doi: 10.1016/j.cam.2012.12.019.  Google Scholar

[24]

T. GudiH. S. Gupta and N. Nataraj, Analysis of an interior penalty method for fourth order problems on polygonal domains, J. Sci. Comput., 54 (2013), 177-199.  doi: 10.1007/s10915-012-9612-9.  Google Scholar

[25]

J. GuoC. WangS. M. Wise and X. Yue, An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation, Commun. Math. Sci., 14 (2016), 489-515.  doi: 10.4310/CMS.2016.v14.n2.a8.  Google Scholar

[26]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. Ⅰ. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.  doi: 10.1137/0719018.  Google Scholar

[27]

R. H. W. Hoppe and C. Linsenmann, $\rm C^0$-interior penalty discontinuous Galerkin approximation of a sixth-order Cahn-Hilliard equation, Contrib. PDEs Appl., 47 (2019), 297-325.   Google Scholar

[28]

S. Kaessmair and P. Steinmann, Comparative computational analysis of the Cahn–Hilliard equation with emphasis on $C^1$-continuous methods, J. Comput. Phys., 322 (2016), 783-803.  doi: 10.1016/j.jcp.2016.07.005.  Google Scholar

[29]

D. Li and Z. Qiao, On second order semi-implicit Fourier spectral methods for 2D Cahn–Hilliard equations, J. Sci. Comput., 70 (2017), 301-341.  doi: 10.1007/s10915-016-0251-4.  Google Scholar

[30]

A. Miranville, The Cahn–Hilliard Equation: Recent Advances and Applications, SIAM, Philadelphia, 2019. doi: 10.1137/1.9781611975925.  Google Scholar

[31]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2$^nd$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar

[32]

G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn–Hilliard equation and its applicability to related energy-based models, Arch. Comput. Methods Eng., 22 (2015), 269-289.  doi: 10.1007/s11831-014-9112-1.  Google Scholar

Figure 1.  Energy dissipation of the CH energy, $ E $, and the discrete energy, $ F $, for three different mesh sizes (and consequently three different time step sizes). All other parameters are defined in the paragraph above
Figure 2.  Evolution towards a steady state. The times are displayed above each image. The space step size is $ h = {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{128}\;} $, the time step size is $ \tau = 0.002h $, $ \varepsilon = 0.0625 $, and all other parameters are defined in the paragraph above
Table 1.  Errors and convergence rates of the C$ ^0 $-IP method. Parameters and initial conditions are given in the text
$ h $ $ \left\| {error_{\phi}} \right\|_{{2,h}} $ rate $ \left\| {error_{\phi}} \right\|_{{H^1}} $ rate
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{8}\;} $ 25.1808 N/A 1.2793 N/A
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{16}\;} $ 8.60978 1.46233 0.42525 1.50418
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{32}\;} $ 2.57074 1.67457 0.11405 1.86430
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{64}\;} $ 0.70546 1.82202 0.02727 2.09046
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{128}\;} $ 0.17818 1.97961 0.00516 2.63819
$ h $ $ \left\| {error_{\phi}} \right\|_{{2,h}} $ rate $ \left\| {error_{\phi}} \right\|_{{H^1}} $ rate
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{8}\;} $ 25.1808 N/A 1.2793 N/A
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{16}\;} $ 8.60978 1.46233 0.42525 1.50418
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{32}\;} $ 2.57074 1.67457 0.11405 1.86430
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{64}\;} $ 0.70546 1.82202 0.02727 2.09046
$ {{}^{\sqrt{2}}\!\!\diagup\!\!{}_{128}\;} $ 0.17818 1.97961 0.00516 2.63819
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