Quadrature by Matched Asymptotic Expansion (QBMAX): Accelerating Evaluation of Layer Potentials with Boundary Layers
Year: 2026
Type: preprint
Abstract: Evaluating layer potentials with rapidly decaying kernels becomes computationally challenging in the presence of boundary layers, as occurs for Helmholtz layer potentials with large imaginary wavenumbers. We introduce Quadrature by Matched Asymptotic Expansion (QBMAX), a high-order method that extends the Quadrature by Expansion (QBX) framework by incorporating the kernel's asymptotic decay behavior directly into the Taylor expansion of the potential. Numerical results in both 2D and 3D show improvements of up to four digits over QBX for the tested large-parameter cases, while the methods can perform similarly for smaller parameters or more strongly curved geometries. A symbolic weighted operation-count model indicates comparable costs for forming the two expansion kernels. The Taylor expansions have the same formal order, while QBMAX reduces the truncation-error constants in the flat-boundary model and generally exhibits less convergence degradation in the reported experiments.
Source: arXiv (Cornell University)
Authors: Lin, Chaoqi, Wei, Xiaoyu, Klöckner, Andreas
Institution: University of Illinois Urbana-Champaign
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G1326603436, Integral Equation Methods for the Efficient Simulation of Nonlinear Interface Phenomena, CAREER: Towards General-Purpose, High-Order Integral Equation Methods for Computer Simulation in Engineering: Analysis, Algorithm Design, and Applications, Elements: Transformation-Based High-Performance Computing in Dynamic Languages, SHF: Small: Collaborative Research: Transform-to-Perform: Languages, Algorithms, and Solvers for Nonlocal Operators
OpenAlex ID: W7212321668