American Institute of Mathematical Sciences

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August  2022, 15(8): 1919-1939. doi: 10.3934/dcdss.2022038

$ C^1 $-VEM for some variants of the Cahn-Hilliard equation: A numerical exploration

Paola F. Antonietti 1,, , Simone Scacchi 2, , Giuseppe Vacca 3, and  Marco Verani 1,

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

Received  December 2021 Published  August 2022 Early access  February 2022

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: Virtual element method, polytopal meshes, fourth order problems, Cahn-Hilliard equation, impainting, parallel computing.
Mathematics Subject Classification: Primary: 65M60; Secondary: 65M99.
Citation: Paola F. Antonietti, Simone Scacchi, Giuseppe Vacca, Marco Verani. $ C^1 $-VEM for some variants of the Cahn-Hilliard equation: A numerical exploration. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1919-1939. doi: 10.3934/dcdss.2022038
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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.

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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]

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Figure 1.  Example of polygonal meshes of the domain $ (0, 1)^2 $ used in the numerical tests
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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
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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
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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 $
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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 $
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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 $
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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
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mesh $ 1/h $ # elements # nodes # DoFs
QUAD 128 16384 16641 49923
TRI 128 56932 28723 86169
CVT 128 16384 32943 98829
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
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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
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