###Hybrid numerical-asymptotic methods for wave scattering problems
Project ID: 2228bd1207 (You will need this ID for your application)
Research Theme: Mathematical Sciences
UCL Lead department: Mathematics
Lead Supervisor: David Hewett
Project Summary:
Acoustic and electromagnetic wave scattering has applications throughout science and technology, including in imaging, telecommunications, and atmospheric physics. The numerical simulation of such scattering is challenging, especially in the high frequency regime where the wavelength of the incident wave is small compared to the dimensions of the scatterer, since conventional finite element or boundary element methods require the number of degrees of freedom in each spatial dimension to scale in proportion to the frequency. At very high frequencies one can appeal to asymptotic methods like geometrical optics and the geometrical theory of diffraction, but such approximations don’t offer controllable accuracy at fixed frequencies. The hybrid numerical-asymptotic (HNA) approach aims to address this issue by building high frequency asymptotic information into the numerical FEM/BEM approximation space via the inclusion of special oscillatory basis functions, so as to create numerical methods which are controllably accurate and computationally feasible, even at high frequencies. This has been achieved for numerous problems (mostly in 2D so far) including scattering by convex and nonconvex polygons and colinear screens. The aim of this project would be to extend the HNA methodology to new scattering configurations, and develop an open-source software package providing fast implementations of HNA methods, which requires the application of fast and accurate methods for oscillatory quadrature. Depending on the student’s expertise and interests, the project would involve some combination of developing new numerical approximation spaces, proving rigorous error bounds, writing software implementing the methods, and applying the methods to derive numerical solutions to challenging high frequency scattering problems.