2018
DOI: 10.1016/j.jcp.2018.03.011
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A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

Abstract: We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite element method (ray-FEM) [34], whose original version can not handle the singularity close to point sources accurately. This pitfall is addressed by combining the ray-FEM, which is used to compute the smooth far-field of the solution accurately, with a high-order asymptotic expansion close to the point source, which is used to properly ca… Show more

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Cited by 13 publications
(11 citation statements)
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“…Regarding (ii): the first such bounds were proved for the exterior Helmholtz equation in [8] (for A variable and n = 1) and [9], (for n variable and A = I). Recent such bounds were proved in [59,38,32,79] (and for problems posed on bounded domains with impedance boundary conditions [23,Chapter 2], [6,11,67,78,38,39]), the renewed interest due to growing interest in the numerical analysis of Helmholtz equation with variable coefficients [24,6,11,67,30,33,25,26,70,39,32,71,50,36,51].…”
Section: Discussion Of the Novelty Of The Bound In Theorem 16mentioning
confidence: 99%
“…Regarding (ii): the first such bounds were proved for the exterior Helmholtz equation in [8] (for A variable and n = 1) and [9], (for n variable and A = I). Recent such bounds were proved in [59,38,32,79] (and for problems posed on bounded domains with impedance boundary conditions [23,Chapter 2], [6,11,67,78,38,39]), the renewed interest due to growing interest in the numerical analysis of Helmholtz equation with variable coefficients [24,6,11,67,30,33,25,26,70,39,32,71,50,36,51].…”
Section: Discussion Of the Novelty Of The Bound In Theorem 16mentioning
confidence: 99%
“…Can other techniques be combined with transmission conditions as a remedy? We emphasize the recent progress of some techniques for solving wave problems: for example, coarse spaces (Conen, Dolean, Krause and Nataf 2014, Bonazzoli, Dolean, Graham, Spence and Tournier 2018, Bootland, Dolean, Jolivet and Tournier 2021a or deflation (Dwarka and Vuik 2020, Dwarka, Tielen, Möller and Vuik 2021, Bootland, Dwarka, Jolivet, Dolean and Vuik 2021b, time domain solvers (Grote and Tang 2019, Grote, Nataf, Tang and Tournier 2020, Appelo, Garcia and Runborg 2020, Stolk 2021, absorption or shifted-Laplace (Gander, Graham and Spence 2015, Graham, Spence and Vainikko 2017, Graham et al 2020, Hocking and Greif 2021, H-matrix (Beams, Gillman and Hewett 2020, Lorca, Beams, Beecroft and Gillman 2021, Liu, Ghysels, Claus and Li 2021, Bonev and Hesthaven 2022, DPG (Petrides and Demkowicz 2021) and high-frequency asymptotic (Lu, Qian and Burridge 2016, Fang, Qian, Zepeda-Núñez and Zhao 2018, Jacobs and Luo 2021. There are plenty of questions still to be answered in the future.…”
Section: Double Sweep Schwarz Methods For the Layered Medium Wave Pro...mentioning
confidence: 99%
“…However, in room acoustics, since the speed of sound is typically a perturbation about a constant value due to small fluctuations in temperature, our simple model here should be reasonably accurate. There are other ways of modeling the amplitude near a point source that have been developed for use with high-order sweeping methods [107,48]. In Section 1.6, we develop reflection and diffraction BCs for the amplitude.…”
Section: The Point Source Amplitudementioning
confidence: 99%