High Order Boundary Element Methods Accelerated by the Fast Multipole Method in NGSolve

Abstract

This thesis develops a Fast Multipole Method (FMM) framework for 3D boundary integral equations and introduces a new numerically stable wall recurrence for translation operators of spherical basis functions. We further derive stable backward recurrences with dynamic rescaling for spherical Bessel and Hankel functions, ensuring robust evaluation of special functions required by the expansions. The framework is integrated with the Galerkin Boundary Element Method (BEM) in the open source library NGSolve, supporting layer potential operators for Laplace, Helmholtz, Maxwell and Lam ́e equations, with a user friendly Python interface. Together, these contributions make high order BEM practical for large scale 3D problems and provide an open source reference implementation for further research and applications.