An adaptive kernel-split quadrature method for parameter-dependent layer potentials

• Published in: Advances in computational mathematics 2022, Vol.48 (2), Article 12
• Main Authors: Fryklund, Fredrik, Klinteberg, Ludvig af, Tornberg, Anna-Karin
• Format: Article
• Language: English
• Subjects: Adaptive sampling; Advances in Computational Integral Equations; Algorithms; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Integral equations; Kernels; Layer potentials; Mathematical analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Modified Helmholtz equation; Modified Stokes equation; Parameter modification; Partial differential equations; Quadratures; Visualization
• ISSN: 1019-7168; 1572-9044; 1572-9044
• DOI: 10.1007/s10444-022-09927-5

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic, and modified Stokes equations. These equations depend on a parameter, denoted α , and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter α , at an increased cost that scales as log α . Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.