Discretisations of sign-definite formulations for the Helmholtz equation

Lead Research Organisation: University of Reading
Department Name: Mathematics and Statistics


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.

Planned Impact

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.


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Description The Helmholtz equation, the subject of this grant, is a main mathematical model of acoustic wave propagation and scattering. The accurate and efficient numerical solution of the Helmholtz equation is a key goal of computational acoustics. Despite large previous international research efforts by engineers, scientist and mathematicians, this goal is still out of reach for configurations where the important wavelengths of the acoustic waves are small compared to some characteristic length of the computational domain.

One standard approach to numerical solution is to use finite element methods. These lead, in the case when the wavelength is small compared to the diameter of the computational domain, to the need to solve very large systems of linear equations, so large that iterative solution methods are essential. These iterative methods converge very slowly, and the design of so-called preconditioners to speed up the convergence is difficult because standard finite element methods lead to linear systems that are so-called "sign indefinite".

The main aim of this proposal was to explore whether there would be significant improvements in solution efficiency by using finite element methods based on the discretisation of a novel variational reformulation of the Helmholtz, which has the property that it is sign-definite (see A. Moiola and E.A. Spence, SIAM Review 56 (2014) 274-312); this property is in turn inherited by finite element discretisations, and the expectation was that this important additional property would lead to improvements in efficiency, either through; a) the achievement of the same simulation accuracy with smaller linear system size; or b) more efficient iterative solution of the positive definite linear system. Unfortunately, the results of the project were negative: neither of these improvements materialised (see G.C. Diwan, A. Moiola, and E.A. Spence, Journal of Computational and Applied Mathematics 352 (2019) 110-131).
Exploitation Route The findings were largely negative, so that it is not clear that these can be usefully taken forwards.
Sectors Aerospace, Defence and Marine,Construction,Digital/Communication/Information Technologies (including Software),Environment,Manufacturing, including Industrial Biotechology,Transport