Discretisations of sign-definite formulations for the Helmholtz equation

涉及声波、电磁波和弹性波的现象在许多科学技术领域普遍存在。他们精确的计算机模拟是医学成像、非侵入性治疗、碳氢化合物勘探、噪声预测、天线和激光等电子设备设计、超声波、雷达和声纳建模的基础。然而，高频波现象的模拟在计算上是非常具有挑战性的。有限元法，最常用的技术，用于近似偏微分方程，在这种情况下表现不佳。其中一个原因是，当模拟波动问题的方程被重新定义为“变分公式”（即设计具体数值方法所需的积分表达式）时，缺乏称为“符号确定性”的数学性质。申请人最近提出了第一个专门针对具有符号确定性的高频声波问题的变分公式。本项目旨在根据这一公式设计新的具体方法。将调查、实施和分析几种不同的方法，并将其性能与竞争方法进行仔细比较。这将导致：(A)高频波问题的方法比现有方法具有更好的数值性能；(B)更好地理解公式的符号确定性和其他理论性质如何影响其计算特征。此外，将符号确定公式扩展到与电磁波和弹性波有关的更具挑战性的问题。

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.