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Electrostatic Boundary Conditions

Electrostatic Boundary Conditions

Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
The surface integral of an electric field is given by Gauss's law in integral form and is related to...

Poisson's And Laplace's Equation

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.

Electrostatic Boundary Conditions in Dielectrics

Electrostatic Boundary Conditions in Dielectrics

When an electric field passes from one homogeneous medium to another, crossing the boundary between the two mediums imparts a discontinuity in the electric field. This results in electrostatic boundary conditions that depend on the type of mediums the field propagates through.
Consider a case where both the mediums across a boundary are two different dielectric materials. Recall that the electric field and electric displacement are proportional and related through the material's permittivity.

Magnetostatic Boundary Conditions

Magnetostatic Boundary Conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...

Maxwell's Equation Of Electromagnetism

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James Clerk Maxwell (1831–1879) was one of the major contributors to physics in the nineteenth century. Although he died young, he made major contributions to the development of the kinetic theory of gases, to the understanding of color vision, and to understanding the nature of Saturn's rings. He is probably best known for having combined existing knowledge on the laws of electricity and magnetism with his insights into a complete overarching electromagnetic theory, which is represented by...

Differential Form of Maxwell's Equations

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James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and Faraday.

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An Adaptive Fast Multipole Boundary Element Method for Poisson-Boltzmann Electrostatics.

Benzhuo Lu¹, Xiaolin Cheng, Jingfang Huang

¹Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China, Center for Molecular Biophysics, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250, Department of Chemistry & Biochemistry, Center for Theoretical Biological Physics, Department of Pharmacology, Howard Hughes Medical Institute, University of California, San Diego, California 92093.

Journal of Chemical Theory and Computation

|June 12, 2009

Summary

This study presents an improved Poisson-Boltzmann solver using adaptive Fast Multipole Method (FMM) and node-patch discretization. This significantly enhances computational efficiency for large biomolecular systems.

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Finite Element Modelling of a Cellular Electric Microenvironment

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Published on: May 18, 2021

Area of Science:

Computational chemistry
Biophysics
Electrostatics

Background:

Numerical solutions to the Poisson-Boltzmann (PB) equation are crucial for understanding electrostatic solvation effects.
Existing PB solvers are computationally intensive, limiting their application to large biomolecular systems.
Previous work introduced a boundary integral equation-based PB solver accelerated by the Fast Multipole Method (FMM).

Purpose of the Study:

To present an updated, more efficient PB solver.
To improve computational performance and memory usage for large-scale simulations.
To make PB calculations feasible for complex biomolecular systems on standard hardware.

Main Methods:

Implementation of an adaptive FMM to accelerate matrix-vector multiplications.
Adoption of a node-patch discretization scheme, reducing the number of unknowns.
Optimization of load balancing for local- and far-field calculations.

Main Results:

The adaptive FMM significantly improves memory usage and calculation speed compared to nonadaptive methods.
The node-patch scheme reduces unknowns by half without compromising accuracy.
The enhanced solver achieves linear (N) complexity in computational cost and memory.

Conclusions:

The updated PB solver offers substantial improvements in performance and efficiency.
This advancement enables the practical simulation of large biomolecular systems, like ribosomes.
The method is now feasible even on typical desktop computers, broadening accessibility.