Close Search Advanced
Journals
Resources
About Us
Open Access

Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
Preview
Add to basket

Year:    2012

Communications in Computational Physics, Vol. 11 (2012), Iss. 1 : pp. 179–214

Abstract

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L∞ estimates. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.081009.130611a

Communications in Computational Physics, Vol. 11 (2012), Iss. 1 : pp. 179–214

Published online:    2012-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    36

Keywords:   

  1. mFES: A Robust Molecular Finite Element Solver for Electrostatic Energy Computations

    Sakalli, I. | Schöberl, J. | Knapp, E. W.

    Journal of Chemical Theory and Computation, Vol. 10 (2014), Iss. 11 P.5095

    https://doi.org/10.1021/ct5005092 [Citations: 16]

  2. A Posteriori Analysis and Efficient Refinement Strategies for the Poisson--Boltzmann Equation

    Chaudhry, Jehanzeb H.

    SIAM Journal on Scientific Computing, Vol. 40 (2018), Iss. 4 P.A2519

    https://doi.org/10.1137/17M1119846 [Citations: 6]

  3. Two‐grid methods for semilinear interface problems

    Holst, Michael | Szypowski, Ryan | Zhu, Yunrong

    Numerical Methods for Partial Differential Equations, Vol. 29 (2013), Iss. 5 P.1729

    https://doi.org/10.1002/num.21774 [Citations: 9]

  4. Multilevel correction adaptive finite element method for semilinear elliptic equation

    Lin, Qun | Xie, Hehu | Xu, Fei

    Applications of Mathematics, Vol. 60 (2015), Iss. 5 P.527

    https://doi.org/10.1007/s10492-015-0110-x [Citations: 11]

  5. Efficient mesh refinement for the Poisson‐Boltzmann equation with boundary elements

    Ramm, Vicente | Chaudhry, Jehanzeb H. | Cooper, Christopher D.

    Journal of Computational Chemistry, Vol. 42 (2021), Iss. 12 P.855

    https://doi.org/10.1002/jcc.26506 [Citations: 1]

  6. Enriched gradient recovery for interface solutions of the Poisson-Boltzmann equation

    Borleske, George | Zhou, Y.C.

    Journal of Computational Physics, Vol. 421 (2020), Iss. P.109725

    https://doi.org/10.1016/j.jcp.2020.109725 [Citations: 1]

  7. On developing stable finite element methods for pseudo-time simulation of biomolecular electrostatics

    Deng, Weishan | Xu, Jin | Zhao, Shan

    Journal of Computational and Applied Mathematics, Vol. 330 (2018), Iss. P.456

    https://doi.org/10.1016/j.cam.2017.09.004 [Citations: 7]

  8. Towards optimal boundary integral formulations of the Poisson–Boltzmann equation for molecular electrostatics

    Search, Stefan D. | Cooper, Christopher D. | van't Wout, Elwin

    Journal of Computational Chemistry, Vol. 43 (2022), Iss. 10 P.674

    https://doi.org/10.1002/jcc.26825 [Citations: 6]

  9. Beyond Poisson–Boltzmann: Numerical Sampling of Charge Density Fluctuations

    Poitevin, Frédéric | Delarue, Marc | Orland, Henri

    The Journal of Physical Chemistry B, Vol. 120 (2016), Iss. 26 P.6270

    https://doi.org/10.1021/acs.jpcb.6b02650 [Citations: 3]

  10. Regularization methods for the Poisson-Boltzmann equation: Comparison and accuracy recovery

    Lee, Arum | Geng, Weihua | Zhao, Shan

    Journal of Computational Physics, Vol. 426 (2021), Iss. P.109958

    https://doi.org/10.1016/j.jcp.2020.109958 [Citations: 9]

  11. A Domain Decomposition Method for the Poisson--Boltzmann Solvation Models

    Quan, Chaoyu | Stamm, Benjamin | Maday, Yvon

    SIAM Journal on Scientific Computing, Vol. 41 (2019), Iss. 2 P.B320

    https://doi.org/10.1137/18M119553X [Citations: 13]

  12. Renormalized charge and dielectric effects in colloidal interactions: a numerical solution of the nonlinear Poisson–Boltzmann equation for unknown boundary conditions

    Schlaich, Alexander | Tyagi, Sandeep | Kesselheim, Stefan | Sega, Marcello | Holm, Christian

    The European Physical Journal E, Vol. 46 (2023), Iss. 9

    https://doi.org/10.1140/epje/s10189-023-00334-2 [Citations: 2]

  13. Metric tensors for the interpolation error and its gradient inLpnorm

    Xie, Hehu | Yin, Xiaobo

    Journal of Computational Physics, Vol. 256 (2014), Iss. P.543

    https://doi.org/10.1016/j.jcp.2013.09.008 [Citations: 3]

  14. A New Approach to Implement Absorbing Boundary Condition in Biomolecular Electrostatics

    Goni, Md. Osman

    IEEE/ACM Transactions on Computational Biology and Bioinformatics, Vol. 10 (2013), Iss. 3 P.799

    https://doi.org/10.1109/TCBB.2013.96 [Citations: 0]

  15. Biomolecular electrostatics—I want your solvation (model)

    Bardhan, Jaydeep P

    Computational Science & Discovery, Vol. 5 (2012), Iss. 1 P.013001

    https://doi.org/10.1088/1749-4699/5/1/013001 [Citations: 32]

  16. Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation

    Aksoylu, Burak | Bond, Stephen D. | Cyr, Eric C. | Holst, Michael

    Journal of Scientific Computing, Vol. 52 (2012), Iss. 1 P.202

    https://doi.org/10.1007/s10915-011-9539-6 [Citations: 10]

  17. A unified model for mechanical deformations of single stranded DNA-microbeam biosensors considering surface charge effects

    Tan, Zouqing | Feng, Yang | Shi, Xiaohao | Zhang, Nenghui

    Applied Mathematical Modelling, Vol. 135 (2024), Iss. P.559

    https://doi.org/10.1016/j.apm.2024.07.006 [Citations: 0]

  18. Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

    Iglesias, José A. | Nakov, Svetoslav

    Journal of Mathematical Analysis and Applications, Vol. 511 (2022), Iss. 1 P.126065

    https://doi.org/10.1016/j.jmaa.2022.126065 [Citations: 1]

  19. ARGOS: An adaptive refinement goal‐oriented solver for the linearized Poisson–Boltzmann equation

    Nakov, Svetoslav | Sobakinskaya, Ekaterina | Renger, Thomas | Kraus, Johannes

    Journal of Computational Chemistry, Vol. 42 (2021), Iss. 26 P.1832

    https://doi.org/10.1002/jcc.26716 [Citations: 2]

  20. Communication: Modeling charge-sign asymmetric solvation free energies with nonlinear boundary conditions

    Bardhan, Jaydeep P. | Knepley, Matthew G.

    The Journal of Chemical Physics, Vol. 141 (2014), Iss. 13

    https://doi.org/10.1063/1.4897324 [Citations: 24]

  21. ddX: Polarizable continuum solvation from small molecules to proteins

    Nottoli, Michele | Herbst, Michael F. | Mikhalev, Aleksandr | Jha, Abhinav | Lipparini, Filippo | Stamm, Benjamin

    WIREs Computational Molecular Science, Vol. 14 (2024), Iss. 4

    https://doi.org/10.1002/wcms.1726 [Citations: 0]

  22. An efficient adaptive multigrid method for the elasticity eigenvalue problem

    Xu, Fei | Huang, Qiumei | Xie, Manting

    BIT Numerical Mathematics, Vol. 62 (2022), Iss. 4 P.2005

    https://doi.org/10.1007/s10543-022-00939-7 [Citations: 0]

  23. Reliable Computer Simulation Methods for Electrostatic Biomolecular Models Based on the Poisson–Boltzmann Equation

    Kraus, Johannes | Nakov, Svetoslav | Repin, Sergey

    Computational Methods in Applied Mathematics, Vol. 20 (2020), Iss. 4 P.643

    https://doi.org/10.1515/cmam-2020-0022 [Citations: 3]

  24. An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

    Yin, Peimeng | Huang, Yunqing | Liu, Hailiang

    Communications in Computational Physics, Vol. 16 (2014), Iss. 2 P.491

    https://doi.org/10.4208/cicp.270713.280214a [Citations: 5]

  25. Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations

    Xu, Fei | Huang, Qiumei | Yang, Huiting | Ma, Hongkun

    Applied Numerical Mathematics, Vol. 172 (2022), Iss. P.224

    https://doi.org/10.1016/j.apnum.2021.10.001 [Citations: 2]

  26. A two-component Matched Interface and Boundary (MIB) regularization for charge singularity in implicit solvation

    Geng, Weihua | Zhao, Shan

    Journal of Computational Physics, Vol. 351 (2017), Iss. P.25

    https://doi.org/10.1016/j.jcp.2017.09.026 [Citations: 25]

  27. An Unfitted Finite Element Poisson–Boltzmann Solver with Automatic Resolving of Curved Molecular Surface

    Liu, Ziyang | Gui, Sheng | Lu, Benzhuo | Zhang, Linbo

    The Journal of Physical Chemistry B, Vol. 128 (2024), Iss. 27 P.6463

    https://doi.org/10.1021/acs.jpcb.4c01894 [Citations: 0]

  28. Coupling finite and boundary element methods to solve the Poisson–Boltzmann equation for electrostatics in molecular solvation

    Bosy, Michał | Scroggs, Matthew W. | Betcke, Timo | Burman, Erik | Cooper, Christopher D.

    Journal of Computational Chemistry, Vol. 45 (2024), Iss. 11 P.787

    https://doi.org/10.1002/jcc.27262 [Citations: 0]

  29. A kernel-free boundary integral method for the nonlinear Poisson-Boltzmann equation

    Zhou, Han | Yang, Jiahe | Ying, Wenjun

    Journal of Computational Physics, Vol. 493 (2023), Iss. P.112423

    https://doi.org/10.1016/j.jcp.2023.112423 [Citations: 1]

  30. SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules

    Jiang, Yi | Xie, Yang | Ying, Jinyong | Xie, Dexuan | Yu, Zeyun

    Computational and Mathematical Biophysics, Vol. 3 (2015), Iss. 1

    https://doi.org/10.1515/mlbmb-2015-0011 [Citations: 3]

  31. Progress in developing Poisson-Boltzmann equation solvers

    Li, Chuan | Li, Lin | Petukh, Marharyta | Alexov, Emil

    Computational and Mathematical Biophysics, Vol. 1 (2013), Iss. 2013 P.42

    https://doi.org/10.2478/mlbmb-2013-0002 [Citations: 24]

  32. On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary

    Wang, Siwen | Alexov, Emil | Zhao, Shan

    Mathematical Biosciences and Engineering, Vol. 18 (2021), Iss. 2 P.1370

    https://doi.org/10.3934/mbe.2021072 [Citations: 6]

  33. Error Estimator Using Higher Order FEM for an Interface Problem

    Randrianarivony, Maharavo

    Applied Mathematics, Vol. 08 (2017), Iss. 12 P.1769

    https://doi.org/10.4236/am.2017.812127 [Citations: 1]

  34. Interface solutions of partial differential equations with point singularity

    Zhou, Y.C. | Gupta, Varun

    Journal of Computational and Applied Mathematics, Vol. 362 (2019), Iss. P.400

    https://doi.org/10.1016/j.cam.2018.10.006 [Citations: 2]

  35. A new box iterative method for a class of nonlinear interface problems with application in solving Poisson–Boltzmann equation

    Xie, Dexuan | Ying, Jinyong

    Journal of Computational and Applied Mathematics, Vol. 307 (2016), Iss. P.319

    https://doi.org/10.1016/j.cam.2016.01.005 [Citations: 10]
  36. Image-Based Geometric Modeling and Mesh Generation

    Finite Element Modeling of Biomolecular Systems in Ionic Solution

    Lu, Benzhuo

    2013

    https://doi.org/10.1007/978-94-007-4255-0_14 [Citations: 2]

  37. A regularization approach for solving the super-Gaussian Poisson-Boltzmann model with heterogeneous dielectric functions

    Wang, Siwen | Shao, Yuanzhen | Alexov, Emil | Zhao, Shan

    Journal of Computational Physics, Vol. 464 (2022), Iss. P.111340

    https://doi.org/10.1016/j.jcp.2022.111340 [Citations: 5]

  38. Convergence of a diffuse interface Poisson-Boltzmann (PB) model to the sharp interface PB model: A unified regularization formulation

    Shao, Yuanzhen | McGowan, Mark | Wang, Siwen | Alexov, Emil | Zhao, Shan

    Applied Mathematics and Computation, Vol. 436 (2023), Iss. P.127501

    https://doi.org/10.1016/j.amc.2022.127501 [Citations: 1]

  39. Dielectric continuum methods for quantum chemistry

    Herbert, John M.

    WIREs Computational Molecular Science, Vol. 11 (2021), Iss. 4

    https://doi.org/10.1002/wcms.1519 [Citations: 101]

  40. Multi-scale fusion network: A new deep learning structure for elliptic interface problems

    Ying, Jinyong | Liu, Jiaxuan | Chen, Jiaxin | Cao, Shen | Hou, Muzhou | Chen, Yinghao

    Applied Mathematical Modelling, Vol. 114 (2023), Iss. P.252

    https://doi.org/10.1016/j.apm.2022.10.006 [Citations: 6]

  41. Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation

    Kraus, Johannes | Nakov, Svetoslav | Repin, Sergey I.

    Computational Methods in Applied Mathematics, Vol. 20 (2020), Iss. 2 P.293

    https://doi.org/10.1515/cmam-2018-0252 [Citations: 3]

  42. A domain decomposition method for the polarizable continuum model based on the solvent excluded surface

    Quan, Chaoyu | Stamm, Benjamin | Maday, Yvon

    Mathematical Models and Methods in Applied Sciences, Vol. 28 (2018), Iss. 07 P.1233

    https://doi.org/10.1142/S0218202518500331 [Citations: 8]

  43. Adaptive hp-Finite Element Computations for Time-Harmonic Maxwell’s Equations

    Jiang, Xue | Zhang, Linbo | Zheng, Weiying

    Communications in Computational Physics, Vol. 13 (2013), Iss. 2 P.559

    https://doi.org/10.4208/cicp.231111.090312a [Citations: 6]