David Krantz

David Krantz, Anna-Karin Tornberg

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.

David Krantz, Ludvig af Klinteberg, Anna-Karin Tornberg

Dual-space multilevel kernel-splitting (DMK) is a fast summation framework that combines ideas from the fast multipole method, Ewald summation, and multilevel summation. Originally formulated for free-space problems, and later extended to fully periodic problems on a cube, it decomposes the kernel interaction into a smooth global contribution and a hierarchy of localized interactions evaluated on an octree. We extend DMK to problems on rectangular cuboids with periodic boundary conditions in one, two, or three coordinate directions. The periodization leverages the fact that interactions on all tree levels below the root are localized, allowing for their evaluation with minimal modification on a cubical tiling of the domain. The remaining smooth root-level far-field contribution is evaluated in Fourier space, with Fourier series in the periodic directions and Fourier integrals in the free directions. For reduced periodicity, truncated kernels are used to regularize singular and near-singular Fourier kernels, yielding rapidly convergent trapezoidal discretizations and a unified treatment of all periodicities. For large-aspect-ratio cuboids, the root-level sum can be accelerated using the fast Fourier transform. We validate the method for the electrostatic potential and Stokeslet, stresslet and rotlet potentials, for all periodicities and a wide range of aspect ratios. Numerical experiments show that the periodization adds only a small overhead to the original free-space DMK algorithm, also for high-aspect-ratio cuboids. The resulting method provides a framework for applying DMK to problems with mixed periodicity on rectangular cuboids, and extends naturally to other non-oscillatory kernels for which a kernel split is available.