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

Paola F. Antonietti; Simone Scacchi; Giuseppe Vacca; Marco Verani

doi:10.3934/dcdss.2022038

Article Contents

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

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$ 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

Published: August 2022

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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.

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

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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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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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