Carlos Pérez Arancibia

Date: 10 November 2021

Time: 12.45 - 12.15 Hours

Room: RA1501 & MS-Teams

Speaker: Carlos Pérez Arancibia


Title: Density Interpolation Methods for Boundary Integral Equations


Boundary integral equation (BIE) methods are powerful techniques to solve linear partial differential equations (PDEs) having a known fundamental solution. They can easily handle unbounded domains and radiation conditions at infinity, they give rise to reduced-size linear systems which, although dense, can be efficiently solved by means of Krylov-subspace solvers in conjunction with the Fast Multipole or H-matrix compression algorithms, and, moreover, in the context of wave propagation problems, they do not suffer from dispersion errors. The practical implementation of these methods, however, entails dealing with challenging singular boundary integrals. 

In this talk, I will present a class of general regularization techniques for the numerical evaluation of weakly singular, singular, hypersingular, and nearly singular boundary integral operators associated with classical PDEs of the elliptic type. The proposed techniques address longstanding efficiency, accuracy, and practical implementation issues that have hindered the applicability of BIE methods in science and engineering. Relying on Green’s third identity and local interpolations of density functions in terms of homogeneous solutions of the underlying PDE, these techniques regularize the singularities present in boundary integral operators, recasting them in terms of integrands that are bounded or even more regular depending on the density interpolation order. The resulting boundary integrals can then be accurately and efficiently evaluated using elementary off-the-shelf quadrature rules. A variety of numerical examples demonstrate the effectiveness of the proposed methodology in the context of Nyström and boundary element methods for the Laplace, Helmholtz, elastostatic, and time-harmonic Maxwell and elastodynamic boundary value problems.