A hybrid method for systems of closely spaced dielectric spheres and ions

We develop an efficient, semianalytical, spectrally accurate, and well-conditioned hybrid method for the study of electrostatic fields in composites consisting of an arbitrary distribution of dielectric spheres and ions that are loosely or densely (close to touching) packed. We first derive a closed-form formula for the image potential of a general multipole source of arbitrary order outside a dielectric sphere. Based on this formula, a hybrid method is then constructed to solve the boundary value problem by combining these analytical methods of image charges and image multipoles with the spectrally accurate mesh-free method of moments. The resulting linear system is well conditioned and requires many fewer unknowns on material interfaces as compared with standard boundary integral equation methods, in which the formulation becomes increasingly ill-conditioned and the number of unknowns also increases sharply as the spheres approach each other or ions approach the spheres due to the geometric and physical stiffness. We further apply the fast multipole method to accelerate the calculation of charge-charge, charge-multipole, and multipole-multipole interactions to achieve optimal computational complexity. The accuracy and efficiency of the scheme are demonstrated via several numerical examples.