CMX Lunch Seminar
Geometric approaches to infinite domain problems
Numerical simulations on infinite domains are challenging. In this talk, we will take geometric approaches to analyze the problems and provide new solutions. One problem we tackle is the perfectly matched layer (PML) problem for computational waves on infinite domains. PML is a theoretical wave-absorbing medium attached to the truncated domain that generates no reflection at the interface. However, over the past 25 years, the method still suffers from numerical reflections due to discretization error. We derive the PML based on principles in discrete differential geometry, and for the first time, we obtain a discrete PML that generates no numerical reflections.
Another geometric approach to infinite domain problems is to study transformations and symmetries at the level of PDEs. It turns out that within this geometry of functions and PDEs, the distinction between the notions of exterior and interior for a domain is no longer prominent. In the talk, we will generalize the Kelvin transformations as a general strategy to compactify PDE problems.