Computing with Singular and Nearly Singular Integrals
Time: October 22, 2007, 1pm - 2pm
Place: D344, LSRC
Speaker: Tom Beale

The solutions of partial differential equations are often expressed as singular integrals, using, for example, the fundamental solution of Laplace's equation. In some problems, quantities of interest are integrals over surfaces in 3D or curves in 2D. Such functions could occur as solutions of boundary value problems or could result from changes in material properties at interfaces, e.g. velocity and pressure in fluid flow or electric and magnetic fields. In a numerical method we often want to use values of these quantities on a regular grid with a separate representation of the interface. If an integral such as a layer potential is evaluated at a point on the interface it is singular, but the value at a grid point off the interface but nearby leads to a nearly singular integral. We will describe a relatively simple approach to computing such integrals. A regularization of the singularity is used to control the discretization error, and corrections can be found analytically for both discretization and regularization.