ABSTRACT:
A colleague of mine once described the Helmholtz equation as “the simplest difficult PDE”. The reason that it is “simple” is that it is a second-order linear elliptic equation. The two main reasons that it is “difficult” are that, when the wavenumber is large, 1) the solutions are highly oscillatory, and 2) the usual variational formulations of Helmholtz boundary value problems are sign-indefinite. In this talk, I will argue that this second difficulty is not an inherent feature of the Helmholtz equation itself, only of its standard formulations. I will do this by presenting new sign-definite formulations of several Helmholtz boundary value problems. This is joint work with Andrea Moiola (Reading).
ADDITIONAL INFORMATION:
Euan’s research interests are at the interface between analysis and numerical analysis of partial differential equations (PDEs); problems involving the scattering of acoustic and electromagnetic waves. http://people.bath.ac.uk/eas25/