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A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs
$ C^1 $-VEM for some variants of the Cahn-Hilliard equation: A numerical exploration
1. | MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy |
2. | Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy |
3. | Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy |
We consider the $ C^1 $-Virtual Element Method (VEM) for the conforming numerical approximation of some variants of the Cahn-Hilliard equation on polygonal meshes. In particular, we focus on the discretization of the advective Cahn-Hilliard problem and the Cahn-Hilliard inpainting problem. We present the numerical approximation and several numerical results to assess the efficacy of the proposed methodology.
Correction: Bari is added after the zip code 70125 in third author’s address. We apologize for any inconvenience this may cause.
References:
[1] |
R. A. Adams, Sobolev Spaces, volume 65 of Pure and Applied Mathematics, Academic Press, New York-London, 1975. |
[2] |
A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani,
A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626.
doi: 10.1002/mma.4548. |
[3] |
B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo,
Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), 376-391.
doi: 10.1016/j.camwa.2013.05.015. |
[4] |
P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent and J. Koster,
A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matr. Anal. Appl., 23 (2001), 15-41.
doi: 10.1137/S0895479899358194. |
[5] |
P. R. Amestoy, A. Guermouche, J.-Y. L'Excellent and S. Pralet,
Hybrid scheduling for the parallel solution of linear systems, Paral. Comput., 32 (2006), 136-156.
doi: 10.1016/j.parco.2005.07.004. |
[6] |
P. F. Antonietti, L. Beirão da Veiga, S. 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. |
[7] |
P. F. Antonietti, G. Manzini, S. Scacchi and M. Verani,
A review on arbitrarily regular virtual element methods for elliptic partial differential equations, Math. Models Methods Appl. Sci., 31 (2021), 2825-2853.
doi: 10.1142/S0218202521500627. |
[8] |
P. F. Antonietti, G. Manzini and M. Verani,
The conforming virtual element method for polyharmonic problems, Comput. Math. Appl., 79 (2020), 2021-2034.
doi: 10.1016/j.camwa.2019.09.022. |
[9] |
J. H. Argyris, I. Fried and D. W. Scharpf,
The TUBA family of plate elements for the matrix displacement method, Aeronaut. J. R. Aeronaut. Soc., 72 (1968), 701-709.
doi: 10.1017/S000192400008489X. |
[10] |
A. C. Aristotelous, O. Karakashian and S. M. Wise,
A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2211-2238.
doi: 10.3934/dcdsb.2013.18.2211. |
[11] |
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang and H. Zhang, PETSc Users Manual, Technical Report ANL-95/11 - Revision 3.14, Argonne National Laboratory, 2020. |
[12] |
K. Bao, Y. Shi, S. Sun and X.-P. Wang,
A finite element method for the numerical solution of the coupled Cahn-Hilliard and navier-stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.
doi: 10.1016/j.jcp.2012.07.027. |
[13] |
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo,
Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.
doi: 10.1142/S0218202512500492. |
[14] |
L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo,
The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci., 24 (2014), 1541-1573.
doi: 10.1142/S021820251440003X. |
[15] |
L. Beirão da Veiga, F. Dassi and A. Russo,
High-order virtual element method on polyhedral meshes, Comput. Math. Appl., 74 (2017), 1110-1122.
doi: 10.1016/j.camwa.2017.03.021. |
[16] |
L. Beirão da Veiga, F. Dassi and A. Russo,
A $C^1$ virtual element method on polyhedral meshes, Comput. Math. Appl., 79 (2020), 1936-1955.
doi: 10.1016/j.camwa.2019.06.019. |
[17] |
L. Beirão da Veiga, C. Lovadina and A. Russo,
Stability analysis for the virtual element method, Math. Models Methods Appl. Sci., 27 (2017), 2557-2594.
doi: 10.1142/S021820251750052X. |
[18] |
L. Beirão da Veiga and G. Manzini,
A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34 (2014), 759-781.
doi: 10.1093/imanum/drt018. |
[19] |
L. Beirão da Veiga and G. Manzini,
Residual a posteriori error estimation for the virtual element method for elliptic problems, ESAIM Math. Model. Numer. Anal., 49 (2015), 577-599.
doi: 10.1051/m2an/2014047. |
[20] |
K. Bell,
A refined triangular plate bending finite element, Int. J. Numer. Meth. Eng., 1 (1969), 101-122.
doi: 10.1002/nme.1620010108. |
[21] |
A. Bertozzi, S. Esedoǧlu and A. Gillette,
Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.
doi: 10.1137/060660631. |
[22] |
A. L. Bertozzi, S. Esedoglu and A. Gillette,
Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.
doi: 10.1109/TIP.2006.887728. |
[23] |
M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. R. Hughes and C. M. Landis,
A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Engrg., 217-220 (2012), 77-95.
doi: 10.1016/j.cma.2012.01.008. |
[24] |
S. C. Brenner, Q. Guan and L.-Y. Sung,
Some estimates for virtual element methods, Comput. Methods Appl. Math., 17 (2017), 553-574.
doi: 10.1515/cmam-2017-0008. |
[25] |
S. C. Brenner and L.-Y. Sung, Virtual enriching operators, Calcolo, 56 (2019), Paper No. 44, 25 pp.
doi: 10.1007/s10092-019-0338-z. |
[26] |
S. C. Brenner and L.-Y. Sung,
Virtual element methods on meshes with small edges or faces, Math. Models Methods Appl. Sci., 28 (2018), 1291-1336.
doi: 10.1142/S0218202518500355. |
[27] |
F. Brezzi and L. D. Marini,
Virtual element method for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), 455-462.
doi: 10.1016/j.cma.2012.09.012. |
[28] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. Ⅰ. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258.
doi: 10.1002/9781118788295.ch4. |
[29] |
C. Chatelain, T. Balois, P. Ciarletta and M. Ben Amar,
Emergence of microstructural patterns in skin cancer: A phase separation analysis in a binary mixture, New Journal of Physics, 13 (2011), 115013.
doi: 10.1088/1367-2630/13/11/115013. |
[30] |
F. Chave, D. A. Di Pietro, F. Marche and F. Pigeonneaux,
A hybrid high-order method for the Cahn-Hilliard problem in mixed form, SIAM J. Numer. Anal., 54 (2016), 1873-1898.
doi: 10.1137/15M1041055. |
[31] |
L.-Q. Chen,
Phase-field models for microstructure evolution, Annual Review of Materials Research, 32 (2002), 113-140.
doi: 10.1146/annurev.matsci.32.112001.132041. |
[32] |
C. Chinosi and L. D. Marini,
Virtual element method for fourth order problems: $L^2$-estimates, Comput. Math. Appl., 72 (2016), 1959-1967.
doi: 10.1016/j.camwa.2016.02.001. |
[33] |
R. W. Clough and J. L. Tocher, editors, Finite Element Stiffness Matrices for Analysis of Plates in Bending, Proceedings of the Conference on Matrix Methods in Structural Mechanics, 1965. |
[34] |
F. Della Porta and M. Grasselli,
Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529-1553.
doi: 10.3934/dcdsb.2015.20.1529. |
[35] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant,
Modeling elasticity in crystal growth, Physical Review Letters, 88 (2002), 2457011-2457014.
doi: 10.1103/PhysRevLett.88.245701. |
[36] |
C. M. Elliott and D. A. French,
A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989), 884-903.
doi: 10.1137/0726049. |
[37] |
C. M. Elliott and D. A. French,
Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128.
doi: 10.1093/imamat/38.2.97. |
[38] |
C. M. Elliott and Z. Songmu,
On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[39] |
H. Emmerich,
Advances of and by phase-field modelling in condensed-matter physics, Advances in Physics, 57 (2008), 1-87.
doi: 10.1080/00018730701822522. |
[40] |
H. Emmerich, L. Gránásy and H. Löwen,
Selected issues of phase-field crystal simulations., European Physical Journal Plus, 126 (2011), 1-18.
|
[41] |
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. 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. |
[42] |
R. D. Falgout and U. M. Yang, Hypre: A library of high performance preconditioners, In P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J. Dongarra, editors, Computational Science — ICCS 2002, pages 632–641, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. |
[43] |
X. Feng,
Fully discrete finite element approximations of the navier-stokes-cahn- hilliard diffuse interface model for two-phase fluid flows, SIAM Journal on Numerical Analysis, 44 (2006), 1049-1072.
doi: 10.1137/050638333. |
[44] |
X. Feng and A. Prohl,
Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84.
doi: 10.1007/s00211-004-0546-5. |
[45] |
F. Frank, C. Liu, A. Scanziani, F. O. Alpak and B. Riviere,
An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods, Journal of Colloid and Interface Science, 523 (2018), 282-291.
doi: 10.1016/j.jcis.2018.02.075. |
[46] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumour growth, European J. Appl. Math., 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[47] |
C. G. Gal and M. Grasselli,
Asymptotic behavior of a Cahn-Hilliard-navier-stokes system in 2d, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[48] |
C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp.
doi: 10.1007/s00526-016-0992-9. |
[49] |
M. Grasselli and M. Pierre,
Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM Math. Model. Numer. Anal., 50 (2016), 1523-1560.
doi: 10.1051/m2an/2015092. |
[50] |
H. Gómez, V. M. Calo, Y. Bazilevs and T. J. R. Hughes,
Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4333-4352.
doi: 10.1016/j.cma.2008.05.003. |
[51] |
A. Hawkins-Daarud, K. G. van der Zee and J. T. Oden,
Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24.
doi: 10.1002/cnm.1467. |
[52] |
V. E. Henson and U. M. Yang,
BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), 155-177.
doi: 10.1016/S0168-9274(01)00115-5. |
[53] |
J. Hu, T. Lin and Q. Wu, A construction of ${C}^r$ conforming finite element spaces in any dimension, 2021. arXiv: 2103.14924. |
[54] |
J. Hu and S. Zhang,
The minimal conforming $H^k$ finite element spaces on $\mathbb{R}^n$ rectangular grids, Math. Comp., 84 (2015), 563-579.
doi: 10.1090/S0025-5718-2014-02871-8. |
[55] |
X. Huang, ${H}^m$-conforming virtual elements in arbitrary dimension, 2021. arXiv: 2105.12973. |
[56] |
D. Kay, V. Styles and E. Süli,
Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM J. Numer. Anal., 47 (2009), 2660-2685.
doi: 10.1137/080726768. |
[57] |
D. Kay, V. Styles and R. Welford,
Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces and Free Boundaries, 10 (2008), 15-43.
doi: 10.4171/IFB/178. |
[58] |
C. Kuhn and R. Müller,
A continuum phase field model for fracture, Engineering Fracture Mechanics, 77 (2010), 3625-3634.
doi: 10.1016/j.engfracmech.2010.08.009. |
[59] |
M. Li, J. Zhao, C. Huang and S. Chen, Conforming and nonconforming VEMS for the fourth-order reaction-subdiffusion equation: A unified framework, IMA J. Numer. Anal., 2021.
doi: 10.1093/imanum/drab030. |
[60] |
C. Liu, F. Frank and B. M. Rivière,
Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn-Hilliard equation, Numer. Methods Partial Differential Equations, 35 (2019), 1509-1537.
doi: 10.1002/num.22362. |
[61] |
J. Liu, L. Dedè, J. A. Evans, M. J. Borden and T. J. R. Hughes,
Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow, J. Comput. Phys., 242 (2013), 321-350.
doi: 10.1016/j.jcp.2013.02.008. |
[62] |
X. Liu, Z. He and Z. Chen, A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed form, Comput. Phys. Commun., 246 (2020), 106870, 11 pp.
doi: 10.1016/j.cpc.2019.106870. |
[63] |
C. Lovadina, D. Mora and I. Velásquez,
A virtual element method for the von Kármán equations, ESAIM Math. Model. Numer. Anal., 55 (2021), 533-560.
doi: 10.1051/m2an/2020085. |
[64] |
C. Miehe, F. Welschinger and M. Hofacker,
Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83 (2010), 1273-1311.
doi: 10.1002/nme.2861. |
[65] |
A. Miranville, The Cahn-Hilliard Equation, volume 95 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. Recent advances and applications.
doi: 10.1137/1.9781611975925. |
[66] |
N. Moelans, B. Blanpain and P. Wollants,
An introduction to phase-field modeling of microstructure evolution, Calphad: Computer Coupling of Phase Diagrams and Thermochemistry, 32 (2008), 268-294.
doi: 10.1016/j.calphad.2007.11.003. |
[67] |
D. Mora, G. Rivera and I. Velásquez,
A virtual element method for the vibration problem of Kirchhoff plates, ESAIM Math. Model. Numer. Anal., 52 (2018), 1437-1456.
doi: 10.1051/m2an/2017041. |
[68] |
D. Mora and A. Silgado, A ${C}^1$ virtual element method for the stationary quasi-geostrophic equations of the ocean, Comput. Math. Appl., 2021. |
[69] |
D. Mora and I. Velásquez,
A virtual element method for the transmission eigenvalue problem, Math. Models Methods Appl. Sci., 28 (2018), 2803-2831.
doi: 10.1142/S0218202518500616. |
[70] |
D. Mora and I. Velásquez, Virtual element for the buckling problem of Kirchhoff-Love plates, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112687, 22 pp.
doi: 10.1016/j.cma.2019.112687. |
[71] |
F. Regazzoni, N. Parolini and M. Verani,
Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers, Comput. Methods Appl. Mech. Engrg., 338 (2018), 562-596.
doi: 10.1016/j.cma.2018.04.035. |
[72] |
I. Steinbach,
Phase-field models in materials science, Modelling and Simulation in Materials Science and Engineering, 17 (2009).
doi: 10.1088/0965-0393/17/7/073001. |
[73] |
E. L. Thomas, D. M. Anderson, C. S. Henkee and D. Hoffman,
Periodic area-minimizing surfaces in block copolymers, Nature, 334 (1988), 598-601.
doi: 10.1038/334598a0. |
[74] |
C. Wang and S. M. Wise,
An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.
doi: 10.1137/090752675. |
[75] |
G. N. Wells, E. Kuhl and K. Garikipati,
A discontinuous Galerkin method for the Cahn-Hilliard equation, J. Comput. Phys., 218 (2006), 860-877.
doi: 10.1016/j.jcp.2006.03.010. |
[76] |
S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini,
Three-dimensional multispecies nonlinear tumor growth-I. Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543.
doi: 10.1016/j.jtbi.2008.03.027. |
[77] |
S. Zhang,
A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math., 59 (2009), 219-233.
doi: 10.1016/j.apnum.2008.02.002. |
[78] |
S. Zhang,
A family of differentiable finite elements on simplicial grids in four space dimensions, Math. Numer. Sin., 38 (2016), 309-324.
|
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, volume 65 of Pure and Applied Mathematics, Academic Press, New York-London, 1975. |
[2] |
A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani,
A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626.
doi: 10.1002/mma.4548. |
[3] |
B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo,
Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), 376-391.
doi: 10.1016/j.camwa.2013.05.015. |
[4] |
P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent and J. Koster,
A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matr. Anal. Appl., 23 (2001), 15-41.
doi: 10.1137/S0895479899358194. |
[5] |
P. R. Amestoy, A. Guermouche, J.-Y. L'Excellent and S. Pralet,
Hybrid scheduling for the parallel solution of linear systems, Paral. Comput., 32 (2006), 136-156.
doi: 10.1016/j.parco.2005.07.004. |
[6] |
P. F. Antonietti, L. Beirão da Veiga, S. 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. |
[7] |
P. F. Antonietti, G. Manzini, S. Scacchi and M. Verani,
A review on arbitrarily regular virtual element methods for elliptic partial differential equations, Math. Models Methods Appl. Sci., 31 (2021), 2825-2853.
doi: 10.1142/S0218202521500627. |
[8] |
P. F. Antonietti, G. Manzini and M. Verani,
The conforming virtual element method for polyharmonic problems, Comput. Math. Appl., 79 (2020), 2021-2034.
doi: 10.1016/j.camwa.2019.09.022. |
[9] |
J. H. Argyris, I. Fried and D. W. Scharpf,
The TUBA family of plate elements for the matrix displacement method, Aeronaut. J. R. Aeronaut. Soc., 72 (1968), 701-709.
doi: 10.1017/S000192400008489X. |
[10] |
A. C. Aristotelous, O. Karakashian and S. M. Wise,
A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2211-2238.
doi: 10.3934/dcdsb.2013.18.2211. |
[11] |
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang and H. Zhang, PETSc Users Manual, Technical Report ANL-95/11 - Revision 3.14, Argonne National Laboratory, 2020. |
[12] |
K. Bao, Y. Shi, S. Sun and X.-P. Wang,
A finite element method for the numerical solution of the coupled Cahn-Hilliard and navier-stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.
doi: 10.1016/j.jcp.2012.07.027. |
[13] |
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo,
Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.
doi: 10.1142/S0218202512500492. |
[14] |
L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo,
The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci., 24 (2014), 1541-1573.
doi: 10.1142/S021820251440003X. |
[15] |
L. Beirão da Veiga, F. Dassi and A. Russo,
High-order virtual element method on polyhedral meshes, Comput. Math. Appl., 74 (2017), 1110-1122.
doi: 10.1016/j.camwa.2017.03.021. |
[16] |
L. Beirão da Veiga, F. Dassi and A. Russo,
A $C^1$ virtual element method on polyhedral meshes, Comput. Math. Appl., 79 (2020), 1936-1955.
doi: 10.1016/j.camwa.2019.06.019. |
[17] |
L. Beirão da Veiga, C. Lovadina and A. Russo,
Stability analysis for the virtual element method, Math. Models Methods Appl. Sci., 27 (2017), 2557-2594.
doi: 10.1142/S021820251750052X. |
[18] |
L. Beirão da Veiga and G. Manzini,
A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34 (2014), 759-781.
doi: 10.1093/imanum/drt018. |
[19] |
L. Beirão da Veiga and G. Manzini,
Residual a posteriori error estimation for the virtual element method for elliptic problems, ESAIM Math. Model. Numer. Anal., 49 (2015), 577-599.
doi: 10.1051/m2an/2014047. |
[20] |
K. Bell,
A refined triangular plate bending finite element, Int. J. Numer. Meth. Eng., 1 (1969), 101-122.
doi: 10.1002/nme.1620010108. |
[21] |
A. Bertozzi, S. Esedoǧlu and A. Gillette,
Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.
doi: 10.1137/060660631. |
[22] |
A. L. Bertozzi, S. Esedoglu and A. Gillette,
Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.
doi: 10.1109/TIP.2006.887728. |
[23] |
M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. R. Hughes and C. M. Landis,
A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Engrg., 217-220 (2012), 77-95.
doi: 10.1016/j.cma.2012.01.008. |
[24] |
S. C. Brenner, Q. Guan and L.-Y. Sung,
Some estimates for virtual element methods, Comput. Methods Appl. Math., 17 (2017), 553-574.
doi: 10.1515/cmam-2017-0008. |
[25] |
S. C. Brenner and L.-Y. Sung, Virtual enriching operators, Calcolo, 56 (2019), Paper No. 44, 25 pp.
doi: 10.1007/s10092-019-0338-z. |
[26] |
S. C. Brenner and L.-Y. Sung,
Virtual element methods on meshes with small edges or faces, Math. Models Methods Appl. Sci., 28 (2018), 1291-1336.
doi: 10.1142/S0218202518500355. |
[27] |
F. Brezzi and L. D. Marini,
Virtual element method for plate bending problems, Comput. Methods Appl. Mech. Engrg., 253 (2013), 455-462.
doi: 10.1016/j.cma.2012.09.012. |
[28] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. Ⅰ. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258.
doi: 10.1002/9781118788295.ch4. |
[29] |
C. Chatelain, T. Balois, P. Ciarletta and M. Ben Amar,
Emergence of microstructural patterns in skin cancer: A phase separation analysis in a binary mixture, New Journal of Physics, 13 (2011), 115013.
doi: 10.1088/1367-2630/13/11/115013. |
[30] |
F. Chave, D. A. Di Pietro, F. Marche and F. Pigeonneaux,
A hybrid high-order method for the Cahn-Hilliard problem in mixed form, SIAM J. Numer. Anal., 54 (2016), 1873-1898.
doi: 10.1137/15M1041055. |
[31] |
L.-Q. Chen,
Phase-field models for microstructure evolution, Annual Review of Materials Research, 32 (2002), 113-140.
doi: 10.1146/annurev.matsci.32.112001.132041. |
[32] |
C. Chinosi and L. D. Marini,
Virtual element method for fourth order problems: $L^2$-estimates, Comput. Math. Appl., 72 (2016), 1959-1967.
doi: 10.1016/j.camwa.2016.02.001. |
[33] |
R. W. Clough and J. L. Tocher, editors, Finite Element Stiffness Matrices for Analysis of Plates in Bending, Proceedings of the Conference on Matrix Methods in Structural Mechanics, 1965. |
[34] |
F. Della Porta and M. Grasselli,
Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529-1553.
doi: 10.3934/dcdsb.2015.20.1529. |
[35] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant,
Modeling elasticity in crystal growth, Physical Review Letters, 88 (2002), 2457011-2457014.
doi: 10.1103/PhysRevLett.88.245701. |
[36] |
C. M. Elliott and D. A. French,
A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989), 884-903.
doi: 10.1137/0726049. |
[37] |
C. M. Elliott and D. A. French,
Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128.
doi: 10.1093/imamat/38.2.97. |
[38] |
C. M. Elliott and Z. Songmu,
On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[39] |
H. Emmerich,
Advances of and by phase-field modelling in condensed-matter physics, Advances in Physics, 57 (2008), 1-87.
doi: 10.1080/00018730701822522. |
[40] |
H. Emmerich, L. Gránásy and H. Löwen,
Selected issues of phase-field crystal simulations., European Physical Journal Plus, 126 (2011), 1-18.
|
[41] |
G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. 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. |
[42] |
R. D. Falgout and U. M. Yang, Hypre: A library of high performance preconditioners, In P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J. Dongarra, editors, Computational Science — ICCS 2002, pages 632–641, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. |
[43] |
X. Feng,
Fully discrete finite element approximations of the navier-stokes-cahn- hilliard diffuse interface model for two-phase fluid flows, SIAM Journal on Numerical Analysis, 44 (2006), 1049-1072.
doi: 10.1137/050638333. |
[44] |
X. Feng and A. Prohl,
Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84.
doi: 10.1007/s00211-004-0546-5. |
[45] |
F. Frank, C. Liu, A. Scanziani, F. O. Alpak and B. Riviere,
An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods, Journal of Colloid and Interface Science, 523 (2018), 282-291.
doi: 10.1016/j.jcis.2018.02.075. |
[46] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumour growth, European J. Appl. Math., 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[47] |
C. G. Gal and M. Grasselli,
Asymptotic behavior of a Cahn-Hilliard-navier-stokes system in 2d, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[48] |
C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp.
doi: 10.1007/s00526-016-0992-9. |
[49] |
M. Grasselli and M. Pierre,
Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM Math. Model. Numer. Anal., 50 (2016), 1523-1560.
doi: 10.1051/m2an/2015092. |
[50] |
H. Gómez, V. M. Calo, Y. Bazilevs and T. J. R. Hughes,
Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4333-4352.
doi: 10.1016/j.cma.2008.05.003. |
[51] |
A. Hawkins-Daarud, K. G. van der Zee and J. T. Oden,
Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24.
doi: 10.1002/cnm.1467. |
[52] |
V. E. Henson and U. M. Yang,
BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), 155-177.
doi: 10.1016/S0168-9274(01)00115-5. |
[53] |
J. Hu, T. Lin and Q. Wu, A construction of ${C}^r$ conforming finite element spaces in any dimension, 2021. arXiv: 2103.14924. |
[54] |
J. Hu and S. Zhang,
The minimal conforming $H^k$ finite element spaces on $\mathbb{R}^n$ rectangular grids, Math. Comp., 84 (2015), 563-579.
doi: 10.1090/S0025-5718-2014-02871-8. |
[55] |
X. Huang, ${H}^m$-conforming virtual elements in arbitrary dimension, 2021. arXiv: 2105.12973. |
[56] |
D. Kay, V. Styles and E. Süli,
Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM J. Numer. Anal., 47 (2009), 2660-2685.
doi: 10.1137/080726768. |
[57] |
D. Kay, V. Styles and R. Welford,
Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces and Free Boundaries, 10 (2008), 15-43.
doi: 10.4171/IFB/178. |
[58] |
C. Kuhn and R. Müller,
A continuum phase field model for fracture, Engineering Fracture Mechanics, 77 (2010), 3625-3634.
doi: 10.1016/j.engfracmech.2010.08.009. |
[59] |
M. Li, J. Zhao, C. Huang and S. Chen, Conforming and nonconforming VEMS for the fourth-order reaction-subdiffusion equation: A unified framework, IMA J. Numer. Anal., 2021.
doi: 10.1093/imanum/drab030. |
[60] |
C. Liu, F. Frank and B. M. Rivière,
Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn-Hilliard equation, Numer. Methods Partial Differential Equations, 35 (2019), 1509-1537.
doi: 10.1002/num.22362. |
[61] |
J. Liu, L. Dedè, J. A. Evans, M. J. Borden and T. J. R. Hughes,
Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow, J. Comput. Phys., 242 (2013), 321-350.
doi: 10.1016/j.jcp.2013.02.008. |
[62] |
X. Liu, Z. He and Z. Chen, A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed form, Comput. Phys. Commun., 246 (2020), 106870, 11 pp.
doi: 10.1016/j.cpc.2019.106870. |
[63] |
C. Lovadina, D. Mora and I. Velásquez,
A virtual element method for the von Kármán equations, ESAIM Math. Model. Numer. Anal., 55 (2021), 533-560.
doi: 10.1051/m2an/2020085. |
[64] |
C. Miehe, F. Welschinger and M. Hofacker,
Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83 (2010), 1273-1311.
doi: 10.1002/nme.2861. |
[65] |
A. Miranville, The Cahn-Hilliard Equation, volume 95 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. Recent advances and applications.
doi: 10.1137/1.9781611975925. |
[66] |
N. Moelans, B. Blanpain and P. Wollants,
An introduction to phase-field modeling of microstructure evolution, Calphad: Computer Coupling of Phase Diagrams and Thermochemistry, 32 (2008), 268-294.
doi: 10.1016/j.calphad.2007.11.003. |
[67] |
D. Mora, G. Rivera and I. Velásquez,
A virtual element method for the vibration problem of Kirchhoff plates, ESAIM Math. Model. Numer. Anal., 52 (2018), 1437-1456.
doi: 10.1051/m2an/2017041. |
[68] |
D. Mora and A. Silgado, A ${C}^1$ virtual element method for the stationary quasi-geostrophic equations of the ocean, Comput. Math. Appl., 2021. |
[69] |
D. Mora and I. Velásquez,
A virtual element method for the transmission eigenvalue problem, Math. Models Methods Appl. Sci., 28 (2018), 2803-2831.
doi: 10.1142/S0218202518500616. |
[70] |
D. Mora and I. Velásquez, Virtual element for the buckling problem of Kirchhoff-Love plates, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112687, 22 pp.
doi: 10.1016/j.cma.2019.112687. |
[71] |
F. Regazzoni, N. Parolini and M. Verani,
Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers, Comput. Methods Appl. Mech. Engrg., 338 (2018), 562-596.
doi: 10.1016/j.cma.2018.04.035. |
[72] |
I. Steinbach,
Phase-field models in materials science, Modelling and Simulation in Materials Science and Engineering, 17 (2009).
doi: 10.1088/0965-0393/17/7/073001. |
[73] |
E. L. Thomas, D. M. Anderson, C. S. Henkee and D. Hoffman,
Periodic area-minimizing surfaces in block copolymers, Nature, 334 (1988), 598-601.
doi: 10.1038/334598a0. |
[74] |
C. Wang and S. M. Wise,
An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.
doi: 10.1137/090752675. |
[75] |
G. N. Wells, E. Kuhl and K. Garikipati,
A discontinuous Galerkin method for the Cahn-Hilliard equation, J. Comput. Phys., 218 (2006), 860-877.
doi: 10.1016/j.jcp.2006.03.010. |
[76] |
S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini,
Three-dimensional multispecies nonlinear tumor growth-I. Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543.
doi: 10.1016/j.jtbi.2008.03.027. |
[77] |
S. Zhang,
A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math., 59 (2009), 219-233.
doi: 10.1016/j.apnum.2008.02.002. |
[78] |
S. Zhang,
A family of differentiable finite elements on simplicial grids in four space dimensions, Math. Numer. Sin., 38 (2016), 309-324.
|




mesh | # elements | # nodes | # DoFs | |
QUAD | 128 | 16384 | 16641 | 49923 |
TRI | 128 | 56932 | 28723 | 86169 |
CVT | 128 | 16384 | 32943 | 98829 |
mesh | # elements | # nodes | # DoFs | |
QUAD | 128 | 16384 | 16641 | 49923 |
TRI | 128 | 56932 | 28723 | 86169 |
CVT | 128 | 16384 | 32943 | 98829 |
Advective Cahn-Hilliard problem, evolution of a cross | |||||||||||
QUAD mesh with 147456 elements, DoFs = 444675 | |||||||||||
Mumps | BJ | GAMG | bAMG | ||||||||
nit | nit | it | nit | it | nit | it | |||||
1 | 2.2 | 67.8 | 2.2 | 14.7 | 15.5 | 2.2 | 11.5 | 27.8 | 2.2 | 13.3 | 36.7 |
2 | 2.2 | 39.4 | 2.2 | 33.9 | 10.9 | 2.2 | 13.8 | 19.4 | 2.2 | 13.4 | 21.1 |
4 | 2.2 | 24.4 | 2.2 | 37.4 | 8.6 | 2.2 | 13.8 | 10.1 | 2.2 | 13.8 | 12.9 |
8 | 2.2 | 21.6 | 2.2 | 45.8 | 11.7 | 2.2 | 14.2 | 12.7 | 2.2 | 14.0 | 13.3 |
16 | 2.2 | 14.3 | 2.2 | 46.2 | 6.4 | 2.2 | 14.4 | 7.3 | 2.2 | 14.0 | 8.1 |
32 | 2.2 | 7.8 | 2.2 | 45.0 | 1.1 | 2.2 | 14.4 | 1.7 | 2.2 | 14.1 | 2.3 |
48 | 2.2 | 7.2 | 2.2 | 43.7 | 0.82 | 2.2 | 14.8 | 1.3 | 2.2 | 14.2 | 1.7 |
CVT mesh with 147456 elements, DoFs = 884814 | |||||||||||
Mumps | BJ | GAMG | bAMG | ||||||||
nit | nit | it | nit | it | nit | it | |||||
1 | OoM | OoM | 2.2 | 32.2 | 44.9 | 2.2 | 18.9 | 88.7 | 2.2 | 21.1 | 135.7 |
2 | 2.2 | 202.1 | 2.2 | 79.6 | 36.1 | 2.2 | 25.6 | 67.8 | 2.2 | 24.1 | 172.2 |
4 | 2.2 | 123.9 | 2.2 | 95.9 | 27.6 | 2.2 | 28.5 | 59.8 | 2.2 | 25.6 | 125.3 |
8 | 2.2 | 85.4 | 2.2 | 107.4 | 23.9 | 2.2 | 30.4 | 45.9 | 2.2 | 26.9 | 74.6 |
16 | 2.2 | 53.2 | 2.2 | 110.6 | 14.6 | 2.2 | 30.7 | 39.6 | 2.2 | 27.2 | 38.1 |
32 | 2.2 | 32.4 | 2.2 | 109.7 | 4.7 | 2.2 | 31.2 | 34.3 | 2.2 | 27.1 | 18.1 |
48 | 2.2 | 27.2 | 2.2 | 108.3 | 3.2 | 2.2 | 30.8 | 30.5 | 2.2 | 27.2 | 12.9 |
Advective Cahn-Hilliard problem, evolution of a cross | |||||||||||
QUAD mesh with 147456 elements, DoFs = 444675 | |||||||||||
Mumps | BJ | GAMG | bAMG | ||||||||
nit | nit | it | nit | it | nit | it | |||||
1 | 2.2 | 67.8 | 2.2 | 14.7 | 15.5 | 2.2 | 11.5 | 27.8 | 2.2 | 13.3 | 36.7 |
2 | 2.2 | 39.4 | 2.2 | 33.9 | 10.9 | 2.2 | 13.8 | 19.4 | 2.2 | 13.4 | 21.1 |
4 | 2.2 | 24.4 | 2.2 | 37.4 | 8.6 | 2.2 | 13.8 | 10.1 | 2.2 | 13.8 | 12.9 |
8 | 2.2 | 21.6 | 2.2 | 45.8 | 11.7 | 2.2 | 14.2 | 12.7 | 2.2 | 14.0 | 13.3 |
16 | 2.2 | 14.3 | 2.2 | 46.2 | 6.4 | 2.2 | 14.4 | 7.3 | 2.2 | 14.0 | 8.1 |
32 | 2.2 | 7.8 | 2.2 | 45.0 | 1.1 | 2.2 | 14.4 | 1.7 | 2.2 | 14.1 | 2.3 |
48 | 2.2 | 7.2 | 2.2 | 43.7 | 0.82 | 2.2 | 14.8 | 1.3 | 2.2 | 14.2 | 1.7 |
CVT mesh with 147456 elements, DoFs = 884814 | |||||||||||
Mumps | BJ | GAMG | bAMG | ||||||||
nit | nit | it | nit | it | nit | it | |||||
1 | OoM | OoM | 2.2 | 32.2 | 44.9 | 2.2 | 18.9 | 88.7 | 2.2 | 21.1 | 135.7 |
2 | 2.2 | 202.1 | 2.2 | 79.6 | 36.1 | 2.2 | 25.6 | 67.8 | 2.2 | 24.1 | 172.2 |
4 | 2.2 | 123.9 | 2.2 | 95.9 | 27.6 | 2.2 | 28.5 | 59.8 | 2.2 | 25.6 | 125.3 |
8 | 2.2 | 85.4 | 2.2 | 107.4 | 23.9 | 2.2 | 30.4 | 45.9 | 2.2 | 26.9 | 74.6 |
16 | 2.2 | 53.2 | 2.2 | 110.6 | 14.6 | 2.2 | 30.7 | 39.6 | 2.2 | 27.2 | 38.1 |
32 | 2.2 | 32.4 | 2.2 | 109.7 | 4.7 | 2.2 | 31.2 | 34.3 | 2.2 | 27.1 | 18.1 |
48 | 2.2 | 27.2 | 2.2 | 108.3 | 3.2 | 2.2 | 30.8 | 30.5 | 2.2 | 27.2 | 12.9 |
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