A fast algorithm with error bounds for Quadrature by Expansion

• Published in: Journal of computational physics Vol. 374; pp. 135 - 162
• Main Authors: Wala, Matt, Klöckner, Andreas
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-12-2018; Elsevier Science Ltd
• Subjects: Algorithms; Computational physics; Cost analysis; Error analysis; Fast algorithms; Fast multipole method; Integral equations; Laplace equation; Laplace transforms; Quadrature; Singular integrals; Smoothness; Quadrature; Singular integrals; Fast multipole method; Integral equations; Fast algorithms
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.05.006

Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansions which are then evaluated to compute the near or on-surface value of the potential. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy.
•A new fast multipole algorithm for Quadrature by Expansion (QBX).•Comprehensive error and cost analysis for the case of the 2D Laplace kernel.•Empirical results match modeled error and scaling behavior.