Abstract

The numerical simulation of the Helmholtz and Maxwell equations play a critical role in chip and antennadesign, radar cross section determination, speaker design, biomedical imaging, wireless communications, and the development of new meta-materials and better waveguides to name a few. In order to enable design by simulation for problems arising in these applications, automatically adaptive solvers which resolve the complexity of the geometry and the input data play a critical role. In two dimensions, this has been made possible through the development of high-order integral equation based solvers which rely on well-conditioned integral representations, efficient quadrature formulas, and coupling to fast multipole methods/fast direct solvers. However, much is still to desired of these solvers in three dimensions (both in terms of their efficiency and accuracy), particularly in the context of enabling automatic adaptivity in complex geometries. In this talk, I will present efficient high-order accurate solvers for solving boundary integral equations in complex three dimensional geometries with focus on the following two issues—- quadrature methods for computing singular integrals on high order meshes surfaces, and a locally corrected quadrature framework for fast multipole accelerated iterative solvers and strong skeletonization based direct solvers.