Applied Mathematics Colloquium **SPECIAL DAY/TIME**

The practical application of integral equation methods requires the evaluation of boundary integrals with singular, weakly singular or nearly singular kernels in complicated domains. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by the construction of corrections to high order non-singular rules for specific kernels, by singularity subtraction/cancellation, or by kernel regularization and asymptotic analysis. We have developed a systematic, high order approach that works for any singularity (including hypersingular kernels), based only the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method.