Modified adaptive quadrature by expansion for Laplace and Helmholtz layer potentials in 2D

An adaptive algorithm based on quadrature by expansion (QBX) is proposed for computing layer potentials at target points near or on a smooth boundary in \(\mathbb{R}^2\). The algorithm can be viewed as major modifications to the two-phase algorithm AQBX, proposed recently by Klinteberg et al. [SIAM Journal on Scientific Computing, 40(3), 2018]. In the modified AQBX (MAQBX), we consider sharper bounds for the involved truncation error. As a result, the involved stopping criteria are met earlier, and the total computational cost is reduced. Moreover, MAQBX is a single-phase algorithm and its structure is far simpler than that of AQBX. It is recommended that QBX (or any version of it) should be applied on a small part of the boundary that is near the target point, and a classical quadrature is applied on the rest of the boundary (this is often referred to as local QBX). We partially show that for Laplace and Helmholtz potentials, parametric symmetricity of the target point with respect to the near part, can improve the convergence of QBX. Based on this observation, we suggest the local MAQBX that is very efficient in practice both for computing layer potentials and for solving boundary integral equations via the Nystr"{o}m scheme.