Wave equation methods for frequency domain problems

In this project we consider wave propagation problems in the frequency domain.

This is a fundamental problem in many branches of science where waves of different kinds are concerned, for instance electromagnetics, acoustics, optics and seismology.

We aim to approximate solutions to frequency domain wave equations like the Helmholtz equation in the medium to high frequency regime in three dimensions.

This leads to large scale problems and simulations must be carried out on parallel high-performance computing platforms. The numerical methods must therefore scale well, while being easy to implement and generally applicable. It is very difficult to meet these requirements with standard iterative methods for Helmholtz equation.

We consider instead iterative methods based on the related time domain wave equation. Algorithms for solving the wave equation are memory lean, easy to parallelize and they scale well. There are also many provably stable and high order accurate methods available. This allows us to solve much bigger problems.

Our main goal is to develop algorithms that speed up the iterations, to obtain a better theoretical understanding of the convergence rate and to apply the methods in applications.