Sharp quadrature error bounds for the nearest-neighbor discretization of the regularized stokeslet boundary integral equation

TL;DR

This paper derives sharp quadrature error bounds for the nearest-neighbor discretization of the regularized stokeslet boundary integral equation, clarifying convergence behavior and implications for numerical stability in biological fluid mechanics simulations.

Contribution

It provides the first detailed analysis of quadrature error bounds for the nearest-neighbor discretization, highlighting the dependence on set separation and regularization parameters.

Findings

Quadrature error is linear in $h_q$ for disjoint sets and quadratic for contained sets.

Condition number is insensitive to $ ext{epsilon}$ for disjoint sets and grows linearly with $ ext{epsilon}$ for contained sets.

Error bounds for the general case are proportional to the sum of individual case errors.

Abstract

The method of regularized stokeslets is a powerful numerical method to solve the Stokes flow equations for problems in biological fluid mechanics. A recent variation of this method incorporates a nearest-neighbor discretization to improve accuracy and efficiency while maintaining the ease-of-implementation of the original meshless method. This method contains three sources of numerical error, the regularization error associated from using the regularized form of the boundary integral equations (with parameter $\varepsilon$), and two sources of discretization error associated with the force and quadrature discretizations (with lengthscales $h_f$ and $h_q$). A key issue to address is the quadrature error: initial work has not fully explained observed numerical convergence phenomena. In the present manuscript we construct sharp quadrature error bounds for the nearest-neighbor discretisation, noting that the error for a single evaluation of the kernel depends on the smallest distance ($\delta$) between these discretization sets. The quadrature error bounds are described for two cases: with disjoint sets ($\delta>0$) being close to linear in $h_q$ and insensitive to $\varepsilon$, and contained sets ($\delta=0$) being quadratic in $h_q$ with inverse dependence on $\varepsilon$. The practical implications of these error bounds are discussed with reference to the condition number of the matrix system for the nearest-neighbor method, with the analysis revealing that the condition number is insensitive to $\varepsilon$ for disjoint sets, and grows linearly with $\varepsilon$ for contained sets. Error bounds for the general case ($\delta\geq 0$) are revealed to be proportional to the sum of the errors for each case.