scholarly journals Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

2017 ◽  
pp. 3-26 ◽  
Author(s):  
Ivan G. Graham ◽  
Euan A. Spence ◽  
Eero Vainikko
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition ◽  
High Frequency

High-frequency uniform asymptotics for the Helmholtz equation in a quarter-plane

2013 ◽  
Vol 84 (1-2) ◽  
pp. 1-15
Author(s):  
Min-Hai Huang ◽  
Yu-Qiu Zhao
Keyword(s):  
Helmholtz Equation ◽  
High Frequency ◽  
Uniform Asymptotics ◽  
Quarter Plane

scholarly journals Thin-layer solutions of the Helmholtz equation

2020 ◽  
pp. 1-15
Author(s):  
J. R. OCKENDON ◽  
R. H. TEW
Keyword(s):  
Wave Propagation ◽  
Thin Layer ◽  
Helmholtz Equation ◽  
Open Problem ◽  
High Frequency ◽  
Mathematical Structure ◽  
Mathematical Physics ◽  
Thin Layers ◽  
Frequency Wave ◽  
High Frequency Wave

This paper gives a brief overview of some configurations in which high-frequency wave propagation modelled by Helmholtz equation gives rise to solutions that vary rapidly across thin layers. The configurations are grouped according to their mathematical structure and tractability and one of them concerns a famous open problem of mathematical physics.

An optimized Schwarz method with relaxation for the Helmholtz equation: the negative impact of overlap

2019 ◽  
Vol 53 (1) ◽  
pp. 249-268
Author(s):  
Yongxiang Liu ◽  
Xuejun Xu
Keyword(s):  
Convergence Rate ◽  
Helmholtz Equation ◽  
Domain Decomposition ◽  
Negative Impact ◽  
Mesh Size ◽  
Careful Analysis ◽  
Numerical Results ◽  
Wave Number ◽  
Convergence Behavior ◽  
Schwarz Method

In this paper we study how the overlapping size influences the convergence rate of an optimized Schwarz domain decomposition (DD) method with relaxation in the two subdomain case for the Helmholtz equation. Through choosing suitable parameters, we find that the convergence rate is independent of the wave number k and mesh size h, but sensitively depends on the overlapping size. Furthermore, by careful analysis, we obtain that the convergence behavior deteriorates with the increase of the overlapping size. Numerical results which confirm our theory are given.

scholarly journals High-frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis

2020 ◽  
Vol 52 (1) ◽  
pp. 845-893 ◽  
Author(s):  
S. N. Chandler-Wilde ◽  
E. A. Spence ◽  
A. Gibbs ◽  
V. P. Smyshlyaev
Keyword(s):  
Numerical Analysis ◽  
Helmholtz Equation ◽  
High Frequency

scholarly journals A robust domain decomposition method for the Helmholtz equation with high wave number

2016 ◽  
Vol 50 (3) ◽  
pp. 921-944 ◽  
Author(s):  
Wenbin Chen ◽  
Yongxiang Liu ◽  
Xuejun Xu
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Mesh Size ◽  
Relaxation Parameter ◽  
Point Of View ◽  
Wave Number ◽  
Theoretical Point ◽  
High Wave Number ◽  
High Wave

In this paper we present a robust Robin−Robin domain decomposition (DD) method for the Helmholtz equation with high wave number. Through choosing suitable Robin parameters on different subdomains and introducing a new relaxation parameter, we prove that the new DD method is robust, which means the convergence rate is independent of the wave number k for kh = constant and the mesh size h for fixed k. To the best of our knowledge, from the theoretical point of view, this is a first attempt to design a robust DD method for the Helmholtz equation with high wave number in the literature. Numerical results which confirm our theory are given.

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