the FFT based algorithm to solve the continuum electrostatic field

In the paper[1], the continuum electrostatic simulation in the ion transport through membrane-spanning nanopores is realized by the implicit-solvent method. To solve the problem, the governing equation (Poisson equation for systems with heterogeneous permittivity) is expressed and the electric field is calculated in its reciprocal space by applying 3D-FFT[2]. The system is considered periodic, and a modified vacuum field outside is defined. The rectangular unit cell is discredited into grid points. By iteratively revise this modified vacuum field, the residual of the electric field at the grid points reach its minimum in the real space. After getting the predefined threshold, the iteration is terminated and the reaction potential is calculated. The potential at any point in the domain is interpolate by its eight surrounding grid points. The accuracy and convergence properties of this proposed algorithm are very well, with an overall speed comparable to a typical finite-difference solver.

[1]C. Peter and G. Hummer, Ion transport through membrane-spanning nanopores studied by molecular dynamics simulation and continuum electrostatics calculations, Biophysical Journal, Volume 89, 2222-2234,2005
[2]C. Peter,W F. Gunsteren, P H. Hunenberger, Solving the poisson equation for solute solvent systems using fast Fourier transforms, Journal of chemical physics, Volume 116, Number17, 2002