Article Contents

2022, Volume 15, Issue 8: 1919-1939. Doi: 10.3934/dcdss.2022038

This issue Previous Article A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs Next Article On the time decay for the MGT-type porosity problems

$ 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

^* Corresponding author: Paola F. Antonietti
^* Corresponding author: Paola F. Antonietti

Received: December 2021

Early access: February 2022

Published: August 2022

Abstract / Introduction Full Text(HTML) Figure(6) / Table(2) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 65M60; Secondary: 65M99.

Citation:

Full Text(HTML)

Figure 1. Example of polygonal meshes of the domain $ (0, 1)^2 $ used in the numerical tests

Download: Full-size image PowerPoint slide

Figure 2. Test 2, evolution of a cross with convection on the unit square $ (0, 1)^2 $, $ \gamma = 1/100 $, $ {\rm Pe} = 100 $. Computed solution $ c_h $ at different time snapshots. The mesh parameters are reported in Table 1

Download: Full-size image PowerPoint slide

Figure 3. Test 3, spinoidal decomposition of a random disk with convection on the unit square $ (0, 1)^2 $, $ \gamma = 1/100 $, $ {\rm Pe} = 100 $. Computed solution $ c_h $ at different time snapshots. The mesh parameters are reported in Table 1

Download: Full-size image PowerPoint slide

Figure 4. Test 4, impainting of a double stripe on the unit square $ (0, 1)^2 $. The mesh parameters are reported in Table 1. Computed solution $ c_h $ at different time snapshots. Left: initial configuration ($ t = 0 $). Middle: final configuration ($ t = T = 0.02 $). Right: final configuration ($ t = T = 0.02 $) without smoothing effects, projecting the solution $ c_h $ to 0.95 if $ c_h>0 $ and to $ -0.95 $ if $ c_k<0 $

Download: Full-size image PowerPoint slide

Figure 5. Test 5, impainting of a cross on the unit square $ (0, 1)^2 $. The mesh parameters are reported in Table 1. Computed solution $ c_h $ at different time snapshots. Left: initial configuration ($ t = 0 $). Middle: final configuration ($ t = T = 0.02 $). Right: final configuration ($ t = T = 0.02 $) without smoothing effects, projecting the solution $ c_h $ to 0.95 if $ c_h>0 $ and to $ -0.95 $ if $ c_h<0 $

Download: Full-size image PowerPoint slide

Figure 6. Test 6, impainting of a circle on the unit square $ (0, 1)^2 $. The mesh parameters are reported in Table 1. Computed solution $ c_h $ at different time snapshots. Left: initial configuration ($ t = 0 $). Middle: final configuration ($ t = T = 0.02 $). Right: final configuration ($ t = T = 0.02 $) without smoothing effects, projecting the solution $ c_h $ to 0.95 if $ c_h>0 $ and to $ -0.95 $ if $ c_h<0 $

Download: Full-size image PowerPoint slide

Table 1. Mesh size parameter $ h $, number of elements, number of nodes and number of degrees of freedom (DoFs) of the polygonal meshes used in the numerical tests

mesh	$ 1/h $	# elements	# nodes	# DoFs
QUAD	128	16384	16641	49923
TRI	128	56932	28723	86169
CVT	128	16384	32943	98829

| Show Table

DownLoad: CSV

Table 2. Strong scaling test on QUAD and CVT meshes, Advective Cahn-Hilliard, evolution of a cross. $ p $ = number of procs; nit = average Newton iterations per time step; it = average GMRES iterations per Newton iteration; $ T_{sol} $ = average CPU time in seconds per time step; OoM = out of memory

Advective Cahn-Hilliard problem, evolution of a cross
QUAD mesh with 147456 elements, DoFs = 444675
$ p $	Mumps		BJ			GAMG			bAMG
	nit	$ T_{sol} $	nit	it	$ T_{sol} $	nit	it	$ T_{sol} $	nit	it	$ T_{sol} $
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
$ p $	Mumps		BJ			GAMG			bAMG
	nit	$ T_{sol} $	nit	it	$ T_{sol} $	nit	it	$ T_{sol} $	nit	it	$ T_{sol} $
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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.