A local target specific quadrature by expansion method for evaluation of layer potentials in 3D
This work improves computational efficiency for boundary integral equation methods in computational physics and engineering, though the improvement is incremental.
The authors develop a target-specific quadrature by expansion (QBX) method for evaluating layer potentials in 3D, reducing the expansion from O(p^2) to O(p) terms at the cost of target-dependent coefficients. The method is validated on Laplace's equation in multiply connected domains, demonstrating high accuracy for singular and nearly singular integrals.
Accurate evaluation of layer potentials is crucial when boundary integral equation methods are used to solve partial differential equations. Quadrature by expansion (QBX) is a recently introduced method that can offer high accuracy for singular and nearly singular integrals, using truncated expansions to locally represent the potential. The QBX method is typically based on a spherical harmonics expansion which when truncated at order $p$ has $O(p^2)$ terms. This expansion can equivalently be written with $p$ terms, however paying the price that the expansion coefficients will depend on the evaluation/target point. Based on this observation, we develop a target specific QBX method, and apply it to Laplace's equation on multiply connected domains. The method is local in that the QBX expansions only involve information from a neighborhood of the target point. An analysis of the truncation error in the QBX expansions is presented, practical parameter choices are discussed and the method is validated and tested on various problems.