Adaptive singularity swap quadrature for near-singular layer potentials on axisymmetric surfaces

When numerically evaluating layer potentials at target points close to the domain boundary, specialized quadrature techniques are required for accuracy because of rapid variations in the integrand.

To efficiently achieve a prescribed error tolerance, we introduce an adaptive quadrature method for smooth axisymmetric surfaces in which all algorithmic choices are determined automatically from the requested error tolerance.

Standard quadrature is used wherever it is sufficient, while a specialized near-quadrature correction is applied only for those target points where additional accuracy is required.

This correction combines singularity swap quadrature in the azimuthal direction with adaptive refinement in the polar direction; on the resulting refined polar grid, either standard quadrature or singularity swap quadrature is used depending on the predicted quadrature error.

The method is coupled to a standard quadrature based on the trapezoidal rule in the azimuthal direction and Gauss--Legendre quadrature in the polar direction, and is activated only when that rule is predicted to be insufficient.

Quadrature and interpolation error predictors are derived using complex analysis and are used to control both activation and refinement.

While each surface is assumed to be axisymmetric, the layer density and the overall geometry need not be, allowing applications to configurations with multiple smooth axisymmetric bodies and patchwise discretizations.

Numerical examples for Laplace, Helmholtz, and Stokes layer potentials demonstrate reliable error control across a range of geometries, including multi-body configurations.