Fundamental solution and integral equation techniques form an attractive methodology for many linear PDE boundary value problems and have received considerable interest as well as demonstrated success due to, to name a few, the superior conditioning properties, geometric flexibility, and dimensionality reduction that they naturally afford. Known associated challenges they bring to the integral equation community include (1) treatment of resulting dense operator matrices (typically handled with fast summation technologies) as well as (2) accurate handling of singularities present in the fundamental solutions (requiring specialized quadratures). Similar challenges arise when it is desired to extend these methods to inhomogeneous problems for the same linear PDEs, as occur in e.g. time-stepping methods for nonlinear PDEs, initial value problems for wave scattering, and quantifying fluid mixing (an application we will highlight in this talk). In this setting, evaluation of a volume potential (consisting of the convolution of a volumetric source with a singular kernel, typically the Green function) is required at many (and potentially at arbitrary) locations in the volume in addition to on the domain boundary. This talk introduces and demonstrates efficient and high-order numerical algorithms for singular volume potential evaluation which are compatible with adaptivity and fast algorithms (e.g. singular and near-singular corrected FMMs), handle complex geometry in an exact fashion, couple to existing and mature meshing software projects, exploit translational invariance and orthogonal polynomials, and treat a variety of singular kernels with complete ease.

All work is joint with Shravan Veerapaneni (Michigan), a substantial amount joint with Hai Zhu (Michigan) and some is joint with Marc Bonnet (POEMS CNRS-INRIA-ENSTA, France). Speaker(s): Thomas Anderson (University of Michigan)