A Fast Isogeometric BEM for the Three Dimensional Laplace- and Helmholtz Problems

TL;DR

This paper introduces a high-order isogeometric boundary element method using NURBS for exact geometry and a fast multipole method for efficient computation, demonstrating high convergence rates for Laplace and Helmholtz problems.

Contribution

It combines isogeometric analysis with a fast multipole method for boundary element problems, enabling accurate and efficient solutions for 3D Laplace and Helmholtz equations.

Findings

High convergence rates achieved with higher order B-spline functions.

Efficient handling of dense matrices via interpolation-based fast multipole method.

Numerical examples demonstrate the method's effectiveness and accuracy.

Abstract

We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract the problems arising due to the dense matrices produced by boundary element methods. By solving Laplace and Helmholtz problems via a single layer approach we show, through a series of numerical examples suitable for easy comparison with other numerical schemes, that one can indeed achieve extremely high rates of convergence of the pointwise potential through the utilisation of higher order B-spline-based ansatz functions.