A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics

ABSTRACT

The need to determine electrostatic fields in domains bounded by molecular surfaces arises in a number of nanotechnology applications including: biomolecule design, carbon nanotube simulation, and molecular electron transport analysis. Molecular surfaces are typically smooth, without the corners common in electrical interconnect problems, but are often so geometrically complicated that numerical evaluation of the associated electrostatic fields is extremely time-consuming. In this paper we describe and demonstrate a meshless spectrally-accurate integral equation method that only requires a description of the molecular surface in the form of a collection of surface points. Our meshless method is a synthesis of techniques, suitably adapted, including: spherical harmonic surface interpolation, spectral-element-like integral equation discretization, integral desingularization via variable transformation, and matrix-implicit iterative matrix solution. The spectral accuracy of this combined method is verified using analytically solvable sphere and ellipsoid problems, and then its accuracy and efficiency is demonstrated numerically by solving capacitance and coupled Poisson/linearized Poisson-Boltzmann problems associated with a commonly used model of a molecule in solution. The results demonstrate that for a tolerance of 10⁻³ this new approach reduces the number of unknowns by as much as two orders of magnitude over the more commonly used flat panel methods.

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INDEX TERMS

Primary Classification:
  J. Computer Applications
  J.3 LIFE AND MEDICAL SCIENCES
      Subjects: Biology and genetics

Additional Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

  I. Computing Methodologies
  I.6 SIMULATION AND MODELING
      I.6.5 Model Development
          Subjects: Modeling methodologies

General Terms:
Algorithms, Design, Performance

Keywords:
Poisson-Boltzmann equation, integral equation, meshless, spectral method