Regularized Single and Double Layer Integrals in 3D Stokes Flow

TL;DR

This paper introduces a high-accuracy numerical method for computing single and double layer integrals in 3D Stokes flow on smooth surfaces, using regularization and analytical corrections to handle singularities efficiently.

Contribution

The authors develop a regularization technique with analytical corrections for singular and nearly singular integrals in 3D Stokes flow, improving accuracy without increasing computational complexity.

Findings

Achieves high-order convergence for integrals on smooth surfaces.

Demonstrates effectiveness of corrections near close surfaces.

Maintains computational efficiency without grid refinement.

Abstract

We present a numerical method for computing the single layer (Stokeslet) and double layer (stresslet) integrals in Stokes flow. The method applies to smooth, closed surfaces in three dimensions, and achieves high accuracy both on and near the surface. The singular Stokeslet and stresslet kernels are regularized and, for the nearly singular case, corrections are added to reduce the regularization error. These corrections are derived analytically for both the Stokeslet and the stresslet using local asymptotic analysis. For the case of evaluating the integrals on the surface, as needed when solving integral equations, we design high order regularizations for both kernels that do not require corrections. This approach is direct in that it does not require grid refinement or special quadrature near the singularity, and therefore does not increase the computational complexity of the overall algorithm. Numerical tests demonstrate the uniform convergence rates for several surfaces in both the singular and near singular cases, as well as the importance of corrections when two surfaces are close to each other.