A fast algorithm for Quadrature by Expansion in three dimensions

• Published in: Journal of computational physics Vol. 388; pp. 655 - 689
• Main Authors: Wala, Matt, Klöckner, Andreas
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-07-2019; Elsevier Science Ltd
• Subjects: Algorithms; Computational physics; Fast algorithms; Fast multipole method; Helmholtz equations; Integral equations; Multipoles; Quadrature; Singular integrals; Three dimensional problems; Quadrature; Singular integrals; Fast multipole method; Three dimensional problems; Integral equations; Fast algorithms
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.03.024

This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a modified version of the Fast Multipole Method (FMM). Our scheme extends a recently developed formulation of the FMM for QBX in two dimensions, which, in that setting, achieves mathematically rigorous error and running time bounds. In addition to generalization to three dimensions, we highlight some algorithmic and mathematical opportunities for improved performance and stability. Lastly, we give numerical evidence supporting the accuracy, performance, and scalability of the algorithm through a series of experiments involving the Laplace and Helmholtz equations.
•A new fast algorithm for Quadrature by Expansion (QBX) in three dimensions.•Comprehensive strategies to control truncation, quadrature, and acceleration error.•Empirical results that confirm expected error and scaling behavior.