Error analysis of virtual element method for the Poisson–Boltzmann equation
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Linghan Huang
Abstract
The Poisson–Boltzmann equation, which incorporates the source of the Dirac distribution, has been widely applied in predicting the electrostatic potential of biomolecular systems in solution. In this paper we discuss and analyse the virtual element method for the Poisson–Boltzmann equation on general polyhedral meshes. Nearly optimal error estimates, approaching the best possible accuracy, are achieved for the virtual element approximation in both the L 2-norm and H 1-norm, even when the solution of the entire domain has low regularity. The efficiency of the virtual element method and the validity of the proposed theoretical prediction are confirmed through numerical experiments conducted on various polyhedral meshes.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12161026
Acknowledgments
The authors thank Jianhua Chen and Yang Liu for their valuable discussions on numerical experiments.
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Research ethics: Not applicable.
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Informed consent: Informed consent was obtained from all individuals included in this study, or their legal guardians or wards.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: In writing, we use ChatGPT to correct grammar and spelling mistakes.
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Competing interests: The authors state no conflict of interest.
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Research funding: Y. Yang was supported by the National Natural Science Foundation of China (Grant No. 12161026), Guangxi Natural Science Foundation (2020GXNSFAA159098). S. Shu was supported by the National Natural Science Foundation of China (Grant No. 12371373) and Science Challenge Project (Grant No. TZ2024009).
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Data availability: The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
[1] F. Fogolari, A. Brigo, and H. Molinari, “The Poisson–Boltzmann equation for biomolecular electrostatics: a tool for structural biology,” J. Mol. Recognit., vol. 15, no. 6, pp. 377–392, 2002. https://doi.org/10.1002/jmr.577.Search in Google Scholar PubMed
[2] B. Lu and J. A. McCammon, “Improved boundary element methods for Poisson–Boltzmann electrostatic potential and force calculations,” J. Chem. Theory Comput., vol. 3, no. 3, pp. 1134–1142, 2007. https://doi.org/10.1021/ct700001x.Search in Google Scholar PubMed
[3] L. Chen, M. J. Holst, and J. Xu, “The finite element approximation of the nonlinear Poisson–Boltzmann equation,” SIAM J. Numer. Anal., vol. 45, no. 6, pp. 2298–2320, 2007. https://doi.org/10.1137/060675514.Search in Google Scholar
[4] M. Mirzadeh, M. Theillard, A. Helgadöttir, D. Boy, and F. Gibou, “An adaptive, finite difference solver for the nonlinear Poisson–Boltzmann equation with applications to biomolecular computations,” Commun. Comput. Phys., vol. 13, no. 1, pp. 150–173, 2013. https://doi.org/10.4208/cicp.290711.181011s.Search in Google Scholar
[5] I. Kwon and D. Y. Kwak, “Discontinuous bubble immersed finite element method for Poisson–Boltzmann equation,” Commun. Comput. Phys., vol. 25, no. 3, pp. 928–946, 2019.10.4208/cicp.OA-2018-0014Search in Google Scholar
[6] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, “Basic principles of virtual element methods,” Math. Model Methods Appl. Sci., vol. 23, no. 1, pp. 199–214, 2013. https://doi.org/10.1142/s0218202512500492.Search in Google Scholar
[7] P. F. Antonietti, G. Manzini, S. Scacchi, and M. Verani, “A review on arbitrarily regular conforming virtual element methods for second-and higher-order elliptic partial differential equations,” Math. Model Methods Appl. Sci., vol. 31, no. 14, pp. 2825–2853, 2021. https://doi.org/10.1142/s0218202521500627.Search in Google Scholar
[8] L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, “Virtual element method for general second-order elliptic problems on polygonal meshes,” Math. Model Methods Appl. Sci., vol. 26, no. 4, pp. 729–750, 2016. https://doi.org/10.1142/s0218202516500160.Search in Google Scholar
[9] A. Cangiani, P. Chatzipantelidis, G. Diwan, and E. H. Georgoulis, “Virtual element method for quasilinear elliptic problems,” IMA J. Numer. Anal., vol. 40, no. 4, pp. 2450–2472, 2020. https://doi.org/10.1093/imanum/drz035.Search in Google Scholar
[10] A. Cangiani, G. Manzini, and O. J. Sutton, “Conforming and nonconforming virtual element methods for elliptic problems,” IMA J. Numer. Anal., vol. 37, no. 3, pp. 1317–1354, 2017.10.1093/imanum/drw036Search in Google Scholar
[11] D. Y. Kwak and H. Park, “A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem,” Numer. Algorithm., vol. 91, no. 1, pp. 449–471, 2022. https://doi.org/10.1007/s11075-022-01269-z.Search in Google Scholar
[12] X. Liu, J. Li, and Z. Chen, “A nonconforming virtual element method for the Stokes problem on general meshes,” Comput. Methods Appl. Mech. Eng., vol. 320, pp. 694–711, 2017, https://doi.org/10.1016/j.cma.2017.03.027.Search in Google Scholar
[13] G. Manzini and A. Mazzia, “A virtual element generalization on polygonal meshes of the Scott–Vogelius finite element method for the 2-d Stokes problem,” arXiv preprint arXiv:2112.13292, 2021.10.2172/1787266Search in Google Scholar
[14] L. Beirão da Veiga, C. Lovadina, and D. Mora, “A virtual element method for elastic and inelastic problems on polytope meshes,” Comput. Methods Appl. Mech. Eng., vol. 295, pp. 327–346, 2015, https://doi.org/10.1016/j.cma.2015.07.013.Search in Google Scholar
[15] V. M. Nguyen-Thanh, X. Zhuang, H. Nguyen-Xuan, T. Rabczuk, and P. Wriggers, “A virtual element method for 2d linear elastic fracture analysis,” Comput. Methods Appl. Mech. Eng., vol. 340, pp. 366–395, 2018, https://doi.org/10.1016/j.cma.2018.05.021.Search in Google Scholar
[16] D. van Huyssteen, F. L. Rivarola, G. Etse, and P. Steinmann, “On mesh refinement procedures for the virtual element method for two-dimensional elastic problems,” Comput. Methods Appl. Mech. Eng., vol. 393, p. 114849, 2022, https://doi.org/10.1016/j.cma.2022.114849.Search in Google Scholar
[17] S. Cao, L. Chen, and R. Guo, “A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh,” Math. Model Methods Appl. Sci., vol. 31, no. 14, pp. 2907–2936, 2021. https://doi.org/10.1142/s0218202521500652.Search in Google Scholar
[18] L. Beirão da Veiga, F. Dassi, G. Manzini, and L. Mascotto, “Virtual elements for Maxwell’s equations,” Comput. Math. Appl., vol. 116, pp. 82–99, 2022, https://doi.org/10.1016/j.camwa.2021.08.019.Search in Google Scholar
[19] F. Brezzi and L. D. Marini, “Virtual element methods for plate bending problems,” Comput. Methods Appl. Mech. Eng., vol. 253, pp. 455–462, 2013, https://doi.org/10.1016/j.cma.2012.09.012.Search in Google Scholar
[20] J. Zhao, S. Chen, and B. Zhang, “The nonconforming virtual element method for plate bending problems,” Math. Model Methods Appl. Sci., vol. 26, no. 9, pp. 1671–1687, 2016. https://doi.org/10.1142/s021820251650041x.Search in Google Scholar
[21] Y. Liu, S. Shu, H. Wei, and Y. Yang, “A virtual element method for the steady-state Poisson–Nernst–Planck equations on polygonal meshes,” Comput. Math. Appl., vol. 102, pp. 95–112, 2021, https://doi.org/10.1016/j.camwa.2021.10.002.Search in Google Scholar
[22] Z. Chen and J. Zou, “Finite element methods and their convergence for elliptic and parabolic interface problems,” Numer. Math., vol. 79, no. 2, pp. 175–202, 1998. https://doi.org/10.1007/s002110050336.Search in Google Scholar
[23] B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo, “Equivalent projectors for virtual element methods,” Comput. Math. Appl., vol. 66, no. 3, pp. 376–391, 2013. https://doi.org/10.1016/j.camwa.2013.05.015.Search in Google Scholar
[24] L. Chen, H. Wei, and M. Wen, “An interface-fitted mesh generator and virtual element methods for elliptic interface problems,” J. Comput. Phys., vol. 334, pp. 327–348, 2017, https://doi.org/10.1016/j.jcp.2017.01.004.Search in Google Scholar
[25] R. Hiptmair, J. Li, and J. Zou, “Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems,” Numer. Math., vol. 122, no. 3, pp. 557–578, 2012. https://doi.org/10.1007/s00211-012-0468-6.Search in Google Scholar
[26] M. Feistauer, “On the finite element approximation of a cascade flow problem,” Numer. Math., vol. 50, no. 6, pp. 655–684, 1986. https://doi.org/10.1007/bf01398378.Search in Google Scholar
[27] L. Beirão da Veiga, F. Dassi, and A. Russo, “High-order virtual element method on polyhedral meshes,” Comput. Math. Appl., vol. 74, no. 5, pp. 1110–1122, 2017. https://doi.org/10.1016/j.camwa.2017.03.021.Search in Google Scholar
[28] S. C. Brenner, L. R. Scott, and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 3, New York, Springer, 2008.10.1007/978-0-387-75934-0Search in Google Scholar
[29] X. Ren and J. Wei, “On a semilinear elliptic equation in R2 when the exponent approaches infinity,” J. Math. Anal. Appl., vol. 189, no. 1, pp. 179–193, 1995. https://doi.org/10.1006/jmaa.1995.1011.Search in Google Scholar
[30] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Philadelphia, SIAM, 2002.10.1137/1.9780898719208Search in Google Scholar
[31] I. Babuška, “The finite element method for elliptic equations with discontinuous coefficients,” Computing, vol. 5, no. 3, pp. 207–213, 1970. https://doi.org/10.1007/bf02248021.Search in Google Scholar
[32] R. Verfürth, “A posteriori error estimates for nonlinear problems. finite element discretizations of elliptic equations,” Math. Comput., vol. 62, no. 206, pp. 445–475, 1994. https://doi.org/10.1090/s0025-5718-1994-1213837-1.Search in Google Scholar
[33] L. C. Evans, “Partial differential equations,” Am. Math. Soc., vol. 19, 2010.10.1090/gsm/019Search in Google Scholar
[34] L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, “The hitchhiker’s guide to the virtual element method,” Math. Model Methods Appl. Sci., vol. 24, no. 8, pp. 1541–1573, 2014. https://doi.org/10.1142/s021820251440003x.Search in Google Scholar
[35] M. J. Holst, The Poisson–Boltzmann Equation: Analysis and Multilevel Numerical Solution, University Park, Citeseer, 1994.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Effective highly accurate time integrators for linear Klein–Gordon equations across the scales
- Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
- Stability and error analysis of a semi-implicit scheme for incompressible flows with variable density and viscosity
- Error analysis of virtual element method for the Poisson–Boltzmann equation
Articles in the same Issue
- Frontmatter
- Effective highly accurate time integrators for linear Klein–Gordon equations across the scales
- Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
- Stability and error analysis of a semi-implicit scheme for incompressible flows with variable density and viscosity
- Error analysis of virtual element method for the Poisson–Boltzmann equation
