Discretisations of sign-definite formulations for the Helmholtz equation

Phenomena involving acoustic, electromagnetic and elastic waves are ubiquitous in numerous scientific and technological areas. Their accurate computer simulation is fundamental in medical imaging, non-invasive therapy, hydrocarbon exploration, noise prediction, design of electronic devices such as antennas and lasers, ultrasound, radar and sonar modelling. However, the simulation of wave phenomena at high frequencies is computationally very challenging. The finite element method, the most common technique used to approximate partial differential equations, performs poorly in this regime. One of the reasons for this is that the equations modelling wave problems, when recast as "variational formulations" (namely the integral expressions needed to devise a concrete numerical method), lack a mathematical property named "sign-definiteness". The applicant recently proposed the first variational formulation specifically for high-frequency acoustic wave problems that enjoys sign-definiteness.

This project aims at devising new concrete methods relying on this formulation. Several different methods will be investigated, implemented, analysed and their performance will be carefully compared with competing methods. This will lead to:

(A) methods for high-frequency wave problems with better numerical performance than existing ones;

(B) a better understanding of how sign-definiteness and other theoretical properties of a formulation affect its computational features.

Moreover, extensions of the sign-definite formulation to more challenging problems related to electromagnetic and elastic waves will be tackled.

The computer simulation of propagation and interaction of acoustic, elastic and electromagnetic waves is crucial for numerous real-life applications. This project will eventually lead to numerical methods for the simulation of wave phenomena at large frequencies, the regime that presents the hardest computational difficulties, with better performance than existing schemes.

These methods may be relevant for a range of widely different societal and economic sectors encompassing healthcare, energy, environment, defence and telecommunications. Some examples of applications that can benefit are fast and accurate medical imaging methods, non-invasive therapies such as high-intensity focussed ultrasound (HIFU) surgery, reliable hydrocarbon reservoir exploration, efficient designs of antennas and lasers, detection and design of low-radar-cross-section military vehicles. A special application in environmental sciences is the effective simulation of electromagnetic scattering in atmospheric ice crystals in clouds, which is a crucial feature in climate modelling. Potential industrial beneficiaries include the Met Office, Schlumberger Gould Research, BAE Systems, DSTL, the Institute of Cancer Research.

Moreover, the project will improve the understanding of some relations existing between boundary value problems, variational formulations, and concrete numerical methods. In turn, this may lead to even more advanced numerical methods for wave or other problems.