Layer potentials - quadrature error estimates and approximation with error control

When numerically solving PDEs reformulated as integral equations, so-called layer potentials must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation of such integrals increases rapidly when the evaluation point approaches the surface and the integrand becomes sharply peaked. Error estimates are needed to determine when the accuracy becomes insufficient, and then, a sufficiently accurate special quadrature method needs to be employed.

In this talk, we discuss how to estimate quadrature errors, building up from simple integrals in one dimension to layer potentials over smooth surfaces in three dimensions. We also discuss a new special quadrature technique for axisymmetric surfaces with error control. The underlying technique is so-called interpolatory semi-analytical quadrature in conjunction with a singularity swap technique. Here, adaptive discretisations and parameters are set automatically given an error tolerance, utilising further quadrature and interpolation error estimates derived for this purpose.