Numerical Methods for PDEs Seminar

SPEAKER:  Carlos Perez-Arancibia  (Institute for Mathematical and Computational Engineering, Pontificia Universidad Catolica De Chile)

TITLE:  Density Interpolation Methods

ABSTRACT: 

In this talk, I will present present recent developments on a class of effective and simple-to-implement methods for the numerical evaluation of boundary integral operators and layer potentials associated to classical PDEs, that address longstanding efficiency, accuracy, and practical implementation issues that have hindered the applicability of boundary integral equation methods in science and engineering.  The proposed techniques rely on the use of Green’s third identity and local Taylor-like interpolations of density functions in terms of homogeneous solutions of the underlaying PDE. These techniques effectively regularize the singularities present in boundary integral operator and layer potentials, recasting them in

terms of integrands that are bounded or even more regular depending on the order of the density interpolation.  The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules.  A variety of numerical examples demonstrate the effectiveness of these techniques in the context of Nyström and boundary element methods for the Laplace, Helmholtz, and Maxwell equations.

This is joint work with Catalin Turc (NJIT), Luiz Faria (INRIA, ENSTA ParisTech), and Constantine Sideris (USC).