Electrostatic fields without singularities

Abstract

The following problems that arise in the computation of electrostatic forces and in the Boundary Element Method are considered. Given two convex interior-disjoint polyhedra in 3-space endowed with a volume charge density which is a polynomial in the Cartesian coordinates of R 3 , compute the Coulomb force acting on them. Given two interior-disjoint polygons in 3-space endowed with a surface charge density which is polynomial in the Cartesian coordinates of R 3 , compute the normal component of the Coulomb force acting on them. For both problems adaptive Gaussian approximation algorithms are given, which, for n Gaussian points, in time O( n ), achieve absolute error O(c -√n ) for a constant c > 1. Such a result improves upon previously known best asymptotic bounds. This result is achieved by blending techniques from integral geometry, computational geometry and numerical analysis. In particular, integral geometry is used in order to represent the forces as integrals whose kernal is free from singularities.