A scalable Ewald-free BIE framework for periodic Stokes flow via hierarchical proxy sums

Abstract: Particulate Stokes flow in confined, periodic geometries underlies a broad class of problems in biophysics, microfluidics, and the rheology of complex fluids. Boundary integral equation (BIE) methods are a natural tool for such problems, but existing periodization schemes rely either on periodic Green's functions, which are restrictive for complex confining geometries, or on free-space schemes that solve auxiliary proxy strengths alongside the surface densities in an extended linear system whose cost scales unfavorably in three dimensions. We present a BIE framework for three-dimensional particulate Stokes flow in periodic pipes with circular cross-sections, wall-bounded doubly-periodic, and triply-periodic geometries that uses only the free-space Green's function and avoids both Ewald summation and the extended linear system. Proxy sources placed on equivalent surfaces of the kernel-independent FMM (KIFMM) form the auxiliary basis, and contributions from far image boxes are captured by a hierarchical proxy sum made absolutely convergent by a net-force-zero compatibility condition. The resulting periodization precomputation depends only on the periodic-box geometry, independent of the kernel and of the surfaces inside the box, and is reused verbatim across the Stokeslet, stresslet, and rotlet. Combined with high-order adaptive surface discretizations, the method achieves high-order accuracy at $\mathcal{O}(N)$ cost with a single layer of image boxes in the near field. Numerical examples on dense polydisperse suspensions with thousands of particles and on flow through complex periodic channels, together with strong and weak scaling studies, demonstrate efficient performance on systems with millions of degrees of freedom on distributed-memory architectures.