Discretisations of sign-definite formulations for the Helmholtz equation

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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.