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

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.Comment: 12 pages, 6 figure

Passively parallel regularized stokeslets

Stokes flow, discussed by G.G. Stokes in 1851, describes many microscopic biological flow phenomena, including cilia-driven transport and flagellar motility; the need to quantify and understand these flows has motivated decades of mathematical and computational research. Regularized stokeslet methods, which have been used and refined over the past twenty years, offer significant advantages in simplicity of implementation, with a recent modification based on nearest-neighbour interpolation providing significant improvements in efficiency and accuracy. Moreover this method can be implemented with the majority of the computation taking place through built-in linear algebra, entailing that state-of-the-art hardware and software developments in the latter, in particular multicore and GPU computing, can be exploited through minimal modifications ('passive parallelism') to existing MATLAB computer code. Hence, and with widely-available GPU hardware, significant improvements in the efficiency of the regularized stokeslet method can be obtained. The approach is demonstrated through computational experiments on three model biological flows: undulatory propulsion of multiple C. Elegans, simulation of progression and transport by multiple sperm in a geometrically confined region, and left-right symmetry breaking particle transport in the ventral node of the mouse embryo. In general an order-of-magnitude improvement in efficiency is observed. This development further widens the complexity of biological flow systems that are accessible without the need for extensive code development or specialist facilities.Comment: 21 pages, 7 figures, submitte

Regularized Stokeslet rings:an efficient method for axisymmetric Stokes flow with application to the growing pollen tube

The method of regularized Stokeslets, based on the divergence-free exact solution to the equations of highly viscous flow due to a spatially-smoothed concentrated force, is widely employed in biological fluid mechanics. Many problems of interest are axisymmetric, motivating the study of the azimuthally-integrated form of the Stokeslet which physically corresponds to a ring of smoothed forces. The regularized fundamental solution for the velocity (single layer potential) and stress (double layer potential) due to an axisymmetric ring of smoothed point forces, the `regularized ringlet', is derived in terms of complete elliptic integrals of the first and second kind. The relative errors in the total drag and surrounding fluid velocity for the resistance problem on the translating, rotating unit sphere, as well as the condition number of the underlying resistance matrix, are calculated; the regularized method is also compared to 3D regularized Stokeslets, and the singular method of fundamental solutions. The velocity of Purcell's toroidal swimmer is calculated; regularized ringlets enable accurate evaluation of surface forces and propulsion speeds for non-slender tori. The benefits of regularization are illustrated by a model of the internal cytosolic fluid velocity profile in the rapidly-growing pollen tube. Actomyosin transport of vesicles in the tube is modelled using forces immersed in the fluid, from which it is found that transport along the central actin bundle is essential for experimentally-observed flow speeds to be attained. The effect of tube growth speed on the internal cytosolic velocity is also considered. For axisymmetric problems, the regularized ringlet method exhibits a comparable accuracy to the method of fundamental solutions whilst also allowing for the placement of forces inside of the fluid domain and having more satisfactory convergence properties.Comment: 36 pages, 19 figures ; edited authors' initial