Tolerance-driven close evaluation of the Stokes double layer potential on axisymmetric surfaces
Abstract
We consider boundary integral methods for Stokes mobility and resistance problems involving smooth axisymmetric particles. A primary numerical challenge is the accurate and efficient evaluation of layer potentials at off-surface points close to particle surfaces. We present a tolerance-driven workflow for evaluating the Stokes double layer potential at such evaluation points (targets) to prescribed accuracy while avoiding unnecessary computational cost. For each target--particle interaction, a fast classifier selects the least costly option estimated to meet the tolerance among standard, upsampled, and special quadrature. Geometry-dependent unit-density error indicators are precomputed and tabulated in reduced cylindrical coordinates, then combined on the fly with a local layer-density modifier, making its cost negligible relative to evaluating the potential. Targets requiring special quadrature are treated using a stabilized version of singularity swap surface quadrature: the periodic azimuthal integral is evaluated first using translated singularity swap quadrature to prevent severe cancellation near the surface, followed by adaptive Gauss--Legendre quadrature in the polar direction guided by error indicators. We integrate this workflow into a boundary integral solver with precomputed quadrature by expansion for on-surface self-interactions and demonstrate the workflow's performance for challenging configurations of spheroidal particles. Numerical results show that target classification is highly accurate. The prescribed tolerance is met for nearly all target--particle interactions and the error remains within a modest factor of the tolerance in the few remaining cases. Although the experiments focus on the Stokes double layer potential for spheroids, the off-surface framework applies to general smooth axisymmetric surfaces and can be easily adapted to other Stokes layer potentials.
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BibTeX
@misc{krantz2026tolerancedriven,
title = {{Tolerance-driven close evaluation of the Stokes double layer potential on axisymmetric surfaces}},
author = {David Krantz and Pritpal Matharu and Anna-Karin Tornberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11778},
eprint = {2609.11778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11778},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tolerance-driven close evaluation of the Stokes double layer potential on axisymmetric surfaces
AU - David Krantz
AU - Pritpal Matharu
AU - Anna-Karin Tornberg
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11778
UR - https://arxiv.org/abs/2609.11778
AB - We consider boundary integral methods for Stokes mobility and resistance problems involving smooth axisymmetric particles. A primary numerical challenge is the accurate and efficient evaluation of layer potentials at off-surface points close to particle surfaces. We present a tolerance-driven workflow for evaluating the Stokes double layer potential at such evaluation points (targets) to prescribed accuracy while avoiding unnecessary computational cost. For each target--particle interaction, a fast classifier selects the least costly option estimated to meet the tolerance among standard, upsampled, and special quadrature. Geometry-dependent unit-density error indicators are precomputed and tabulated in reduced cylindrical coordinates, then combined on the fly with a local layer-density modifier, making its cost negligible relative to evaluating the potential. Targets requiring special quadrature are treated using a stabilized version of singularity swap surface quadrature: the periodic azimuthal integral is evaluated first using translated singularity swap quadrature to prevent severe cancellation near the surface, followed by adaptive Gauss--Legendre quadrature in the polar direction guided by error indicators. We integrate this workflow into a boundary integral solver with precomputed quadrature by expansion for on-surface self-interactions and demonstrate the workflow's performance for challenging configurations of spheroidal particles. Numerical results show that target classification is highly accurate. The prescribed tolerance is met for nearly all target--particle interactions and the error remains within a modest factor of the tolerance in the few remaining cases. Although the experiments focus on the Stokes double layer potential for spheroids, the off-surface framework applies to general smooth axisymmetric surfaces and can be easily adapted to other Stokes layer potentials.
ER -