An integral equation formulation of the N-body dielectric spheres problem. Part II: complexity analysis

Abstract

This article is the second in a series of two papers concerning the mathematical study of a boundary integral equation of the second kind that describes the interaction of N dielectric spherical particles undergoing mutual polarisation. The first article presented the numerical analysis of the Galerkin method used to solve this boundary integral equation and derived N-independent convergence rates for the induced surface charges and total electrostatic energy. The current article will focus on computational aspects of the algorithm. We provide a convergence analysis of the iterative method used to solve the underlying linear system and show that the number of liner solver iterations required to obtain a solution is independent of N. Additionally, we present two linear scaling solution strategies for the computation of the approximate induced surface charges. Finally, we consider a series of numerical experiments designed to validate our theoretical results and explore the dependence of the numerical errors and computational cost of solving the underlying linear system on different system parameters.

Manipulating Interactions between Dielectric Particles with Electric Fields: A General Electrostatic Many-Body Framework

“…PDEs similar to the transmission problem (eq ) have previously been considered in the literature (see, e.g., refs and ); however, the key novelty of eq is the addition of contributions due to an external electric field and the presence of point charges on the surface of dielectric particles. These additional terms require significant modifications to earlier definitions − of the electrostatic force and interaction energy for the N -body charged dielectric spheres, and they present additional challenges in the efficient numerical implementation.…”

“…The Galerkin nature of the method we present here allows for a precise mathematical analysis, in terms of accuracy, with respect to and complexity with respect to N , which was previously discussed in Hassan et al − and also included the detailed description of the linear scaling of the method and the accuracy of predictions for the electrostatic energy and forces . However, the model is limited to the assumptions made at the beginning of Section , namely, it does not account for the presence of surface point charge and the effect of an external electric field.…”

“…Since the DtN map is a purely local operator (diagonal in the basis of local spherical harmonics), the solution matrix A does not need to be explicitly computed and stored, and its action on an arbitrary vector can be calculated with linear scaling cost. Further details on the FMM implementation can be found in Lindgren et al 42 Once the vector has been computed and the procedure for applying the solution matrix A to an arbitrary vector in the approximation space is set up, the linear system ( eq 11 ) can be solved using a Krylov subspace solver such as GMRES (see Bramas et al 44 for a detailed convergence analysis of GMRES, as applied to this linear system).…”

“…A mathematically well-founded approach to this problem has been proposed by Lindgren et al, which formulates the many-body electrostatic problem, in terms of a BIE of the second type and uses a spectral Galerkin approximation to solve the resulting equations. This mathematical formalism allows for a rigorous convergence and complexity analysis of the induced surface charge, electrostatic interaction energy, and net forces acting on each particle (see refs − ), since the continuous solution and the Galerkin approximation are both well characterized. In the same contribution, it is mathematically proven that (i) the proposed method scales linearly in computational cost, with respect to the number of particles, and (ii) the approximation error does not degrade as the system size increases.…”