33 research outputs found
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 ), and two sources
of discretization error associated with the force and quadrature
discretizations (with lengthscales and ). 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 ()
between these discretization sets. The quadrature error bounds are described
for two cases: with disjoint sets () being close to linear in
and insensitive to , and contained sets () being
quadratic in with inverse dependence on . 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
for disjoint sets, and grows linearly with for contained sets.
Error bounds for the general case () are revealed to be
proportional to the sum of the errors for each case.Comment: 12 pages, 6 figure