Quadrature and Fast Summation for Boundary Integral Methods

QC 2026-09-16

Abstract

This thesis develops accurate, stable, and efficient algorithms for two challenges in boundary integral methods: numerical integration, or quadrature, of nearly singular integrals and fast summation of interactions among many pairwise interactions. Boundary integral methods reformulate linear elliptic partial differential equations as integral equations on the boundary, reducing the problem dimension and allowing high-order discretizations on smooth geometries. Applications include electrostatics, fixed-frequency wave scattering, and the motion of rigid particles suspended in viscous fluids.

The boundary integrals representing the solution, known as layer potentials, are difficult to evaluate at points on or close to the boundary because their integrands are singular on the boundary and sharply peaked nearby. Standard quadrature loses accuracy as the evaluation point, or target, approaches the boundary, while maintaining accuracy by increasing the number of quadrature points becomes costly. For sufficiently close evaluation, special quadrature is therefore needed. We develop such a method for smooth axisymmetric surfaces, with parameters selected automatically to meet a prescribed quadrature error tolerance, and apply it to layer potentials associated with the Laplace, Helmholtz, and Stokes equations underlying the applications above. For certain strongly singular integrals, large terms may nearly cancel during very close evaluation and cause a severe loss of precision. We analyze this effect and develop a stable reformulation that does not incur significant additional computational cost.

Building on these developments, we construct a tolerance-driven quadrature workflow for a boundary integral solver for Stokes flow around rigid particles. For each target–particle interaction, precomputed error indicators select the least costly quadrature treatment predicted to meet the tolerance, using the stabilized special quadrature only where needed.

The second challenge arises when standard quadrature is applied at many targets, since the contribution from every boundary quadrature point must be included for each target. With N boundary quadrature points and a comparable number of targets, direct summation requires O(N²) work. Fast summation methods reduce this cost by avoiding the explicit evaluation of every pairwise interaction. We extend one such adaptive method, known as dual-space multilevel kernel splitting, to interactions among points in rectangular cuboids of arbitrary aspect ratio, with periodic boundary conditions in one, two, or three coordinate directions. For fixed accuracy, the method reduces the computational cost to O(N). We apply the method to interaction sums arising from the Laplace and Stokes equations.

Together, these methods form a framework in which interactions requiring special treatment are identified and evaluated accurately and stably, while fast summation accelerates the evaluation of the remaining interactions. They provide building blocks for reliable and efficient large-scale simulations of rigid-particle suspensions with close interactions and periodic boundary conditions, including studies of collective particle motion and suspension rheology. The quadrature methods may also support simulations of fibers and filaments in cellular-scale biophysics, while the fast summation method applies more broadly to long-range interaction problems, including electrostatic calculations in molecular dynamics.

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