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A new box iterative method for a class of nonlinear interface problems with application in solving Poisson-Boltzmann equation. (English) Zbl 1348.65170

Summary: In this paper, a new box iterative method for solving a class of nonlinear interface problems is proposed by intermixing linear and nonlinear boundary value problems based on a special seven-overlapped-boxes partition. It is then applied to the construction of a new finite element and finite difference hybrid scheme for solving the Poisson-Boltzmann equation (PBE) – a second order nonlinear elliptic interface problem for computing electrostatics of an ionic solvated protein. Furthermore, a modified Newton minimization algorithm accelerated by a multigrid preconditioned conjugate gradient method is presented to efficiently solve each involved nonlinear boundary value problem. In addition, the analytical solution of a Poisson dielectric test model with a spherical solute region containing multiple charges is expressed in a simple series of Legendre polynomials, resulting in a new PBE test model that works for a large number of point charges. The new PBE hybrid solver is programmed as a software package, and numerically validated on the new PBE test model with 892 point charges. It is also compared to a commonly used finite difference scheme in the accuracy of computing solution and electrostatic free energy for three proteins with up to 2124 atomic charges. Numerical results on six proteins demonstrate its high performance in comparison to the PBE finite element program package reported by Xie (2014).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35J60 Nonlinear elliptic equations
35Q20 Boltzmann equations
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
78A30 Electro- and magnetostatics
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
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[1] Schwarz, H. A., Ueber einige abbildungsaufgaben, J. Reine Angew. Math., 70, 105-120 (1869) · JFM 02.0626.01
[3] Mathew, T. P.A., (Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 61 (2008), Springer Verlag) · Zbl 1147.65101
[4] Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford University Press · Zbl 0931.65118
[5] Toselli, A.; Widlund, O., Domain Decomposition Methods: Algorithms and Theory, Vol. 3 (2005), Springer
[6] Fogolari, F.; Brigo, A.; Molinari, H., The Poisson-Boltzmann equation for biomolecular electrostatics: A tool for structural biology, J. Mol. Recognit., 15, 6, 377-392 (2002)
[7] Honig, B.; Nicholls, A., Classical electrostatics in biology and chemistry, Science, 268, 1144-1149 (1995)
[8] Lu, B.; Zhou, Y.; Holst, M.; McCammon, J., Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications, Commun. Comput. Phys., 3, 5, 973-1009 (2008) · Zbl 1186.92005
[9] Neves-Petersen, M.; Petersen, S., Protein electrostatics: A review of the equations and methods used to model electrostatic equations in biomolecules—Applications in biotechnology, Biotechnol. Annu. Rev., 9, 315-395 (2003)
[10] Ren, P.; Chun, J.; Thomas, D. G.; Schnieders, M. J.; Marucho, M.; Zhang, J.; Baker, N. A., Biomolecular electrostatics and solvation: A computational perspective, Q. Rev. Biophys., 45, 04, 427-491 (2012)
[11] Xiao, L.; Wang, C.; Luo, R., Recent progress in adapting Poisson-Boltzmann methods to molecular simulations, J. Theoret. Comput. Chem., 13, 03, Article 1430001 pp. (2014)
[12] Babuška, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5, 207-213 (1970) · Zbl 0199.50603
[13] Bramble, J.; King, J., A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math., 6, 1, 109-138 (1996) · Zbl 0868.65081
[14] Chen, Z.; Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79, 175-202 (1998) · Zbl 0909.65085
[15] Lamichhane, B. P.; Wohlmuth, B. I., Mortar finite elements for interface problems, Computing, 72, 333-348 (2004) · Zbl 1055.65129
[16] Xie, D.; Zhou, S., A new minimization protocol for solving nonlinear Poisson-Boltzmann mortar finite element equation, BIT, 47, 853-871 (2007) · Zbl 1147.65096
[17] Brandt, A., Multi-level adaptive solutions to boundary value problems, Math. Comp., 31, 138, 333-390 (1977) · Zbl 0373.65054
[18] Olshanskii, M. A.; Tyrtyshnikov, E. E., Iterative Methods for Linear Systems: Theory and Applications (2014), SIAM · Zbl 1320.65050
[19] Deuflhard, P., Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Vol. 35 (2011), Springer Science & Business Media
[20] Knoll, D.; Rider, W., A multigrid preconditioned Newton-Krylov method, SIAM J. Sci. Comput., 21, 2, 691-710 (1999) · Zbl 0952.65102
[21] Mu, L.; Wang, J.; Wei, G.; Ye, X.; Zhao, S., Weak Galerkin methods for second order elliptic interface problems, J. Comput. Phys., 250, 106-125 (2013) · Zbl 1349.65472
[22] Li, Z.; Ito, K., The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (2006), SIAM: SIAM Philadelphia
[23] Peskin, C., The immersed boundary method, Acta Numer., 11, 1, 479-517 (2002) · Zbl 1123.74309
[24] Xia, K.; Wei, G.-W., A Galerkin formulation of the MIB method for three dimensional elliptic interface problems, Comput. Math. Appl., 68, 7, 719-745 (2014) · Zbl 1362.65130
[25] Xia, K.; Zhan, M.; Wei, G.-W., MIB Galerkin method for elliptic interface problems, J. Comput. Appl. Math., 272, 195-220 (2014) · Zbl 1294.65103
[26] Hellrung, J. L.; Wang, L.; Sifakis, E.; Teran, J. M., A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions, J. Comput. Phys., 231, 4, 2015-2048 (2012) · Zbl 1408.65078
[27] Lin, T.; Lin, Y.; Zhang, X., Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal., 53, 2, 1121-1144 (2015) · Zbl 1316.65104
[28] Ying, J.; Xie, D., A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule, J. Comput. Phys., 298, 636-651 (2015) · Zbl 1349.78103
[29] Xie, D., New solution decomposition and minimization schemes for Poisson-Boltzmann equation in calculation of biomolecular electrostatics, J. Comput. Phys., 275, 294-309 (2014) · Zbl 1349.78077
[30] Born, M. Z., Volumen und hydratationswärme der ionen, Z. Phys., 1, 1, 45-48 (1920)
[31] Holst, M.; McCammon, J. A.; Yu, Z.; Zhou, Y.; Zhu, Y., Adaptive finite element modeling techniques for the Poisson-Boltzmann equation, Commun. Comput. Phys., 11, 1, 179-214 (2012) · Zbl 1373.82077
[32] Kirkwood, J. G., Theory of solutions of molecules containing widely separated charges with special application to zwitterions, J. Chem. Phys., 2, 351 (1934) · Zbl 0009.27504
[33] Geng, W.; Yu, S.; Wei, G., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 127, 11, Article 114106 pp. (2007)
[35] Nicholls, A.; Honig, B., A rapid finite difference algorithm, utilizing successive over-relaxation to solve the Poisson-Boltzmann equation, J. Comput. Chem., 12, 435-445 (1991)
[36] Klapper, I.; Hagstrom, R.; Fine, R.; Sharp, K.; Honig, B., Focusing of electric fields in the active site of Cu-Zn superoxide dismutase: Effects of ionic strength and amino-acid modification, Proteins: Struct. Funct. Bioinform., 1, 1, 47-59 (1986)
[37] Rocchia, W.; Alexov, E.; Honig, B., Extending the applicability of the nonlinear Poisson-Boltzmann equation: Multiple dielectric constants and multivalent ions, J. Phys. Chem. B, 105, 6507-6514 (2001)
[38] Jo, S.; Vargyas, M.; Vasko-Szedlar, J.; Roux, B.; Im, W., PBEQ-solver for online visualization of electrostatic potential of biomolecules, Nucleic Acids Res., 36, suppl 2, W270-W275 (2008)
[39] Geng, W.; Yu, S.; Wei, G., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 127, 11, Article 114106 pp. (2007)
[40] Wang, C.; Wang, J.; Cai, Q.; Li, Z.; Zhao, H.-K.; Luo, R., Exploring accurate Poisson-Boltzmann methods for biomolecular simulations, Comput. Theoret. Chem., 1024, 34-44 (2013)
[41] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev., 34, 4, 581-613 (1992) · Zbl 0788.65037
[42] Xu, J.; Zou, J., Some nonoverlapping domain decomposition methods, SIAM Rev., 40, 4, 857-914 (1998) · Zbl 0913.65115
[43] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (2008), Springer-Verlag · Zbl 1135.65042
[44] Jackson, J. D., Classical Electrodynamics (1999), Wiley: Wiley New York · Zbl 0920.00012
[45] Jiang, Y.; Ying, J.; Xie, D., A Poisson-Boltzmann equation test model for protein in spherical solute region and its applications, Mol. Based Math. Biol., 2, 86-97 (2014), open Access · Zbl 1347.92006
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