Title: Adaptive quadrature by expansion for layer potential evaluation in two dimensions,
Abstract: 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.