Adaptive boundary element methods for the computation of the electrostatic capacity on complex polyhedra

TL;DR

This paper introduces an adaptive boundary element method combining error estimation and preconditioning to accurately compute the electrostatic capacity of complex polyhedra, achieving five or more digits of precision.

Contribution

It presents a novel algorithm that enhances the accuracy and efficiency of capacity computation for complex polyhedra using adaptive boundary element techniques.

Findings

Achieves capacity computation with at least 5 digits of accuracy.

Provides implementation codes based on Bempp for practical use.

Discusses adaptive solver techniques for complex geometries.

Abstract

The accurate computation of the electrostatic capacity of three dimensional objects is a fascinating benchmark problem with a long and rich history. In particular, the capacity of the unit cube has widely been studied, and recent advances allow to compute its capacity to more than ten digits of accuracy. However, the accurate computation of the capacity for general three dimensional polyhedra is still an open problem. In this paper, we propose a new algorithm based on a combination of ZZ-type a posteriori error estimation and effective operator preconditioned boundary integral formulations to easily compute the capacity of complex three dimensional polyhedra to 5 digits and more. While this paper focuses on the capacity as a benchmark problem, it also discusses implementational issues of adaptive boundary element solvers, and we provide codes based on the boundary element package Bempp to make the underlying techniques accessible to a wide range of practical problems.