CSCDR seminar with Min Hyung Cho (UMass Lowell), "Accurate Numerical Methods for Acoustic and Electromagnetic Scattering Problems”
Textiles 105A
https://www.cscdr.umassd.edu/seminars
Abstract:
In this talk, I will present Quadrature by Two Expansions (QB2X) for the evaluation of Helmholtz layer potentials, as well as recent progress in developing Green’s function-based numerical methods for Maxwell’s equations. One of the main numerical challenges with these integral equation methods is the accurate and efficient evaluation of layer potentials with singular or nearly-singular kernels, especially when the target point is close to or on the boundary of the domain. By utilizing the smoothness properties of layer potentials in the interior and exterior of the domain, along with special partial-wave or local expansions that automatically satisfy the underlying partial differential equations, the Quadrature by Expansion (QBX) scheme is introduced. In addition to local expansions, the QB2X method employs plane-wave type expansions to achieve a representation that is numerically accurate at all target points within the computational box. Compared to QBX, QB2X incorporates explicit nonlinearity from the boundary geometry into the plane-wave expansions and follows standard Fast Multipole Method (FMM) error analysis. Consequently, QB2X overcomes many of the challenges associated with QBX and is well-suited for complex geometries. If time permits, as a second topic, I will present the Method of Fundamental Solutions (MFS) with a periodization scheme for time-harmonic Maxwell’s equations in bi-periodic multilayered media. This method is robust at the Wood anomaly and can achieve spectral convergence close to machine precision. Work on extending this method to integral equations using a fast direct solver is currently ongoing.