Adaptive quadrature by expansion for layer potential evaluation in two dimensions
For practitioners using boundary integral equation methods, AQBX reduces the difficulty of parameter tuning in QBX, making it more practical for solving PDEs.
The paper introduces AQBX, an adaptive quadrature by expansion method for evaluating layer potentials in 2D, which automates parameter selection to meet a target error tolerance. The scheme achieves accuracy within an order of magnitude of the tolerance for Laplace and Helmholtz equations, simplifying the original QBX approach.
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.