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.Suche 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.Suche 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.Suche 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.Suche 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-2018-0014Suche 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.Suche 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.Suche 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.Suche 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.Suche 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/drw036Suche 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.Suche 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.Suche 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, 2021.10.2172/1787266Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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.Suche 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, 2008.10.1007/978-0-387-75934-0Suche 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.Suche in Google Scholar
[30] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Philadelphia, SIAM, 2002.10.1137/1.9780898719208Suche 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.Suche 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.Suche in Google Scholar
[33] L. C. Evans, “Partial differential equations,” Am. Math. Soc., vol. 19, 2010.10.1090/gsm/019Suche 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.Suche in Google Scholar
[35] M. J. Holst, The Poisson–Boltzmann Equation: Analysis and Multilevel Numerical Solution, University Park, Citeseer, 1994.Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- 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
Artikel in diesem Heft
- 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
