Optimal $ H^1 $-norm error analysis of an IMEX BDF2 nonconforming virtual element scheme for the Allen-Cahn equation

Peizhen Wang; Wang Liu; Jinyang Xia; Peizhen Wang; Wang Liu; Jinyang Xia

doi:10.3934/math.20261103

AIMS Mathematics

2026, Volume 11, Issue 8: 27563-27584. doi: 10.3934/math.20261103

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Research article

Optimal $ H^1 $-norm error analysis of an IMEX BDF2 nonconforming virtual element scheme for the Allen-Cahn equation

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China

Received: 26 May 2026 Revised: 07 August 2026 Accepted: 24 August 2026 Published: 31 August 2026
MSC : 65M06, 65M12

We developed a fully discrete scheme for the Allen-Cahn equation on polygonal meshes by combining a second-order implicit-explicit (IMEX) backward differentiation formula (BDF2) with a nonconforming virtual element method for spatial discretization. To rigorously justify the global Lipschitz condition required in the analysis, we employed an explicitly defined truncated polynomial potential that coincided with the standard quartic double-well potential on $ [-M, M] $ and had quadratic growth outside this interval. Under a suitable time-step restriction, the proposed scheme satisfied a modified discrete energy dissipation law. Optimal error estimates of order $ O(h^{k+1}+\tau^2) $ in the $ L^2 $-norm and $ O(h^k+\tau^2) $ in the $ H^1 $-seminorm were established. In particular, a discrete Laplace operator adapted to the nonconforming virtual element space was introduced to avoid temporal order reduction in the $ H^1 $-error analysis. Numerical experiments confirmed the theoretical convergence rates.
- Allen-Cahn equation,
- nonconforming virtual element method,
- IMEX BDF2,
- modified energy stability,
- optimal $ H^1 $-norm error estimate
Citation: Peizhen Wang, Wang Liu, Jinyang Xia. Optimal $ H^1 $-norm error analysis of an IMEX BDF2 nonconforming virtual element scheme for the Allen-Cahn equation[J]. AIMS Mathematics, 2026, 11(8): 27563-27584. doi: 10.3934/math.20261103

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Abstract

We developed a fully discrete scheme for the Allen-Cahn equation on polygonal meshes by combining a second-order implicit-explicit (IMEX) backward differentiation formula (BDF2) with a nonconforming virtual element method for spatial discretization. To rigorously justify the global Lipschitz condition required in the analysis, we employed an explicitly defined truncated polynomial potential that coincided with the standard quartic double-well potential on $ [-M, M] $ and had quadratic growth outside this interval. Under a suitable time-step restriction, the proposed scheme satisfied a modified discrete energy dissipation law. Optimal error estimates of order $ O(h^{k+1}+\tau^2) $ in the $ L^2 $-norm and $ O(h^k+\tau^2) $ in the $ H^1 $-seminorm were established. In particular, a discrete Laplace operator adapted to the nonconforming virtual element space was introduced to avoid temporal order reduction in the $ H^1 $-error analysis. Numerical experiments confirmed the theoretical convergence rates.

References

[1]	S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085–1095. https://doi.org/10.1016/0001-6160(79)90196-2 doi: 10.1016/0001-6160(79)90196-2
[2]	J. Shen, X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669–1691. https://doi.org/10.3934/dcds.2010.28.1669 doi: 10.3934/dcds.2010.28.1669
[3]	S. Orizaga, G. G. Parra, L. Forman, J. Villegas-Villanueva, Solving Allen-Cahn equations with periodic and nonperiodic boundary conditions using mimetic finite-difference operators, Appl. Math. Comput., 484 (2025), 128993. https://doi.org/10.1016/j.amc.2024.128993 doi: 10.1016/j.amc.2024.128993
[4]	C. Lee, Y. Choi, J. Kim, An explicit stable finite difference method for the Allen-Cahn equation, Appl. Numer. Math., 182 (2022), 87–99. https://doi.org/10.1016/j.apnum.2022.08.006 doi: 10.1016/j.apnum.2022.08.006
[5]	Z. Q. Tan, C. J. Zhang, The discrete maximum principle and energy stability of a new second-order difference scheme for Allen-Cahn equations, Appl. Numer. Math., 166 (2021), 227–237. https://doi.org/10.1016/j.apnum.2021.04.010 doi: 10.1016/j.apnum.2021.04.010
[6]	Q. Du, L. L. Ju, X. Li, Z. H. Qiao, Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation, SIAM J. Numer. Anal., 57 (2019), 875–898. https://doi.org/10.1137/18M118236X doi: 10.1137/18M118236X
[7]	P. Manorot, B. Wongsaijai, K. Poochinapan, Performance of a new stabilized structure-preserving finite element method for the Allen-Cahn equation, Math. Comput. Model. Dyn. Syst., 30 (2024), 972–1008. https://doi.org/10.1080/13873954.2024.2421835 doi: 10.1080/13873954.2024.2421835
[8]	J. Yang, N. Y. Yi, H. Zhang, High-order unconditionally maximum-principle preserving finite element method for the Allen-Cahn equation, Appl. Numer. Math., 188 (2023), 42–61. https://doi.org/10.1016/j.apnum.2023.03.002 doi: 10.1016/j.apnum.2023.03.002
[9]	G. Akrivis, B. Li, Error estimates for fully discrete BDF finite element approximations of the Allen-Cahn equation, IMA J. Numer. Anal., 42 (2022), 363–391. https://doi.org/10.1093/imanum/draa065 doi: 10.1093/imanum/draa065
[10]	Y. Liu, X. Q. Shen, Z. Guan, Y. F. Nie, The adaptive SAV weak Galerkin finite element method for the Allen-Cahn equation, Comput. Math. Appl., 151 (2023), 449–460. https://doi.org/10.1016/j.camwa.2023.10.023 doi: 10.1016/j.camwa.2023.10.023
[11]	X. P. Wang, F. Z. Gao, J. T. Cui, Z. J. Sun, A weak Galerkin finite element method for the Allen-Cahn equation with a nonuniform two-step backward differentiation formula scheme, Numer. Methods Partial Differ. Equations, 39 (2023), 2227–2243. https://doi.org/10.1002/num.22964 doi: 10.1002/num.22964
[12]	R. Li, Y. L. Gao, Z. X. Chen, Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes, Numer. Algorithms, 95 (2024), 1981–2014. https://doi.org/10.1007/s11075-023-01635-5 doi: 10.1007/s11075-023-01635-5
[13]	L. B. da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199–214. https://doi.org/10.1142/S0218202512500492 doi: 10.1142/S0218202512500492
[14]	G. Vacca, L. B. da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differ. Equations, 31 (2015), 2110–2134. https://doi.org/10.1002/num.21982 doi: 10.1002/num.21982
[15]	B. J. Zhang, M. F. Feng, Virtual element method for two-dimensional linear elasticity problems in a mixed weakly symmetric formulation, Appl. Math. Comput., 328 (2018), 1–25. https://doi.org/10.1016/j.amc.2018.01.023 doi: 10.1016/j.amc.2018.01.023
[16]	L. B. da Veiga, F. Dassi, G. Vacca, The Stokes complex for virtual elements in three dimensions, Math. Models Methods Appl. Sci., 30 (2020), 477–512. https://doi.org/10.1142/S0218202520500128 doi: 10.1142/S0218202520500128
[17]	L. B. da Veiga, F. Dassi, G. Manzini, L. Mascotto, Virtual elements for Maxwell's equations, Comput. Math. Appl., 116 (2022), 82–99. https://doi.org/10.1016/j.camwa.2021.08.019 doi: 10.1016/j.camwa.2021.08.019
[18]	Y. P. Chen, Q. L. Gu, J. Huang, Step-by-step solving virtual element schemes based on the scalar auxiliary variable approach with relaxation for Allen-Cahn-type gradient flows, Math. Models Methods Appl. Sci., 34 (2024), 2141–2167. https://doi.org/10.1142/S0218202524500453 doi: 10.1142/S0218202524500453
[19]	P. F. Antonietti, L. B. da Veiga, S. Scacchi, M. Verani, A $C^1$ virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54 (2016), 34–56. https://doi.org/10.1137/15M1008117 doi: 10.1137/15M1008117
[20]	B. A. de Dios, K. Lipnikov, G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal., 50 (2016), 879–904. https://doi.org/10.1051/m2an/2015090 doi: 10.1051/m2an/2015090
[21]	A. Cangiani, V. Gyrya, G. Manzini, The nonconforming virtual element method for the Stokes equations, SIAM J. Numer. Anal., 54 (2016), 3411–3435. https://doi.org/10.1137/15M1049531 doi: 10.1137/15M1049531
[22]	X. Liu, Z. X. Chen, The nonconforming virtual element method for the Navier-Stokes equations, Adv. Comput. Math., 45 (2019), 51–74. https://doi.org/10.1007/s10444-018-9602-z doi: 10.1007/s10444-018-9602-z
[23]	B. Zhang, J. K. Zhao, M. Li, The divergence-free nonconforming virtual element method for the Navier-Stokes problem, Numer. Methods Partial Differ. Equations, 39 (2023), 1977–1995. https://doi.org/10.1002/num.22812 doi: 10.1002/num.22812
[24]	J. K. Zhao, B. Zhang, S. P. Mao, S. C. Chen, The nonconforming virtual element method for the Darcy-Stokes problem, Comput. Methods Appl. Mech. Eng., 370 (2020), 113251. https://doi.org/10.1016/j.cma.2020.113251 doi: 10.1016/j.cma.2020.113251
[25]	M. Li, L. L. Wang, N. Wang, Variable-time-step BDF2 nonconforming VEM for coupled Ginzburg-Landau equations, Appl. Numer. Math., 186 (2023), 378–410. https://doi.org/10.1016/j.apnum.2023.01.022 doi: 10.1016/j.apnum.2023.01.022
[26]	J. K. Zhao, S. C. Chen, B. Zhang, The nonconforming virtual element method for plate-bending problems, Math. Models Methods Appl. Sci., 26 (2016), 1671–1687. https://doi.org/10.1142/S021820251650041X doi: 10.1142/S021820251650041X
[27]	J. K. Zhao, T. L. Wang, B. Zhang, The stabilized nonconforming virtual element method for linear elasticity problems, J. Sci. Comput., 92 (2022), 31. https://doi.org/10.1007/s10915-022-01927-3 doi: 10.1007/s10915-022-01927-3
[28]	A. Dedner, A. Hodson, A higher-order nonconforming virtual element method for the Cahn-Hilliard equation, J. Sci. Comput., 101 (2024), 81. https://doi.org/10.1007/s10915-024-02721-z doi: 10.1007/s10915-024-02721-z
[29]	B. A. de Dios, K. Lipnikov, G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal., 50 (2016), 879–904. https://doi.org/10.1051/m2an/2015090 doi: 10.1051/m2an/2015090
[30]	J. K. Zhao, B. Zhang, X. P. Zhu, The nonconforming virtual element method for parabolic problems, Appl. Numer. Math., 143 (2019), 97–111. https://doi.org/10.1016/j.apnum.2019.04.002 doi: 10.1016/j.apnum.2019.04.002
[31]	X. L. Feng, H. L. Song, T. Tang, J. Yang, Nonlinear stability of the implicit–explicit methods for the Allen-Cahn equation, Inverse Probl. Imaging, 7 (2013), 679–695. https://doi.org/10.3934/ipi.2013.7.679 doi: 10.3934/ipi.2013.7.679
[32]	D. Adak, E. Natarajan, S. Kumar, Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes, Numer. Methods Partial Differ. Equations, 35 (2019), 222–245. https://doi.org/10.1002/num.22298 doi: 10.1002/num.22298

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