David Krantz: Accurate quadrature in boundary integral methods

Abstract:

Boundary integral methods are powerful tools for solving elliptic partial differential equations, as they reduce the problem to an integral equation defined on the domain boundary. A key feature of these methods is the representation of the solution as layer potentials, which involve a convolution of a Green's function with a so-called layer "density" function defined on the boundary. Evaluating these layer potentials near the boundary poses significant numerical challenges due to the Green's function being evaluated close to a singularity. This presentation provides a concise introduction to boundary integral methods, focusing on the challenges of accurately evaluating layer potentials. We will explore the types of singularities commonly encountered in two- and three-dimensional problems and introduce specialized quadrature techniques developed to compute the corresponding layer potentials to high accuracy.