10,085 research outputs found
Adaptive quadrature by expansion for layer potential evaluation in two dimensions
When solving partial differential equations using boundary integral equation
methods, accurate evaluation of singular and nearly singular integrals in layer
potentials is crucial. A recent scheme for this is quadrature by expansion
(QBX), which solves the problem by locally approximating the potential using a
local expansion centered at some distance from the source boundary. In this
paper we introduce an extension of the QBX scheme in 2D denoted AQBX - adaptive
quadrature by expansion - which combines QBX with an algorithm for automated
selection of parameters, based on a target error tolerance. A key component in
this algorithm is the ability to accurately estimate the numerical errors in
the coefficients of the expansion. Combining previous results for flat panels
with a procedure for taking the panel shape into account, we derive such error
estimates for arbitrarily shaped boundaries in 2D that are discretized using
panel-based Gauss-Legendre quadrature. Applying our scheme to numerical
solutions of Dirichlet problems for the Laplace and Helmholtz equations, and
also for solving these equations, we find that the scheme is able to satisfy a
given target tolerance to within an order of magnitude, making it useful for
practical applications. This represents a significant simplification over the
original QBX algorithm, in which choosing a good set of parameters can be hard
Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries
Boundary integral methods are highly suited for problems with complicated
geometries, but require special quadrature methods to accurately compute the
singular and nearly singular layer potentials that appear in them. This paper
presents a boundary integral method that can be used to study the motion of
rigid particles in three-dimensional periodic Stokes flow with confining walls.
A centrepiece of our method is the highly accurate special quadrature method,
which is based on a combination of upsampled quadrature and quadrature by
expansion (QBX), accelerated using a precomputation scheme. The method is
demonstrated for rodlike and spheroidal particles, with the confining geometry
given by a pipe or a pair of flat walls. A parameter selection strategy for the
special quadrature method is presented and tested. Periodic interactions are
computed using the Spectral Ewald (SE) fast summation method, which allows our
method to run in O(n log n) time for n grid points, assuming the number of
geometrical objects grows while the grid point concentration is kept fixed.Comment: 46 pages, 41 figure
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