By Joris Roos

The Department of Mathematics & Statistics invites you to attend a colloquium talk delivered by Abinand Gopal (Yale University).

The talk will be delivered in person and some refreshments will be served. Everyone is welcome.
It is also possible to attend via Zoom.

Title: A new solver for PDEs in exteriors of open arcs

Date: Wednesday, Oct 18
Time: 11 a.m. - Noon
Room: Southwick 313

Abstract: When numerically solving a constant-coefficient elliptic PDE, it is often advantageous to first reformulate the problem as a boundary integral equation. Typically, this is done by expressing the solution as the integral of an unknown density function multiplied by a kernel given by the free space fundamental solution of the PDE or its normal derivative. The unknown density function is then obtained by solving a second kind Fredholm integral equation where the right-hand side is given by the boundary data.

However, when the domain consists of the exterior of an open arc in 2D or an open surface in 3D, these standard integral representations encounter challenges. In this talk, we introduce a new representation which involves the composition of the standard single layer potential operator with a certain hypersingular integral operator. We show that the kernel of the composite operator can be efficiently evaluated numerically and that the resulting discretization can be rapidly inverted using a fast direct solver.his is joint work with Shidong Jiang (Flatiron Institute) and Vladimir Rokhlin (Yale University).