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A Bivariate Spline Solution to the Exterior Helmholtz Equation and Its Applications [article]

Shelvean Kapita, Ming Jun Lai
2018 arXiv   pre-print
We explain how to use smooth bivariate splines of arbitrary degree to solve the exterior Helmholtz equation based on a Perfectly Matched Layer (PML) technique.  ...  The PML weights are also approximated by using spline functions. The computational algorithm developed in [26] is used to approximate the solution of the resulting weighted Helmholtz equation.  ...  Conclusion We have shown how to approximate the solution of the exterior Helmholtz equation by using bivariate splines with a PML layer.  ... 
arXiv:1811.07833v1 fatcat:xwylqq63kvbtvkd5rqojnbghuy

Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations

Tomáš Dohnal, Thomas Hagstrom
2007 Journal of Computational Physics  
Extending the general approach for first-order hyperbolic systems developed in [5], we construct PML equations for the mixed-type system governing propagation of optical wave packets in both 1D and 2D  ...  We prove that the layer equations are stable, absorbing, and perfectly matched.  ...  The construction builds on an existing approach for general first order hyperbolic systems.  ... 
doi:10.1016/j.jcp.2006.10.002 fatcat:ngk4u2hvzjcnpc66lwbo4wuln4

Numerical solution of a one-dimensional nonlocal Helmholtz equation by Perfectly Matched Layers [article]

Yu Du, Jiwei Zhang
2021 arXiv   pre-print
We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML  ...  We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form.  ...  Here we also want to study the solution properties of the nonlocal Helmholtz equation (1.1) and the nonlocal PML equation (2.9).  ... 
arXiv:2007.11193v2 fatcat:zo2v75c5hje45gx7nwwatzqjta

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

Radu Cimpeanu, Anton Martinsson, Matthias Heil
2015 Journal of Computational Physics  
This paper presents a parameter-free perfectly matched layer (PML) method for the finite-element-based solution of the Helmholtz equation. We employ one of Bermúdez et al.'  ...  We characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by  ...  We are grateful to Phil Cotterill and Robert Harter for their comments on a draft of this manuscript. We also wish to thank the referees for their constructive comments and suggestions.  ... 
doi:10.1016/j.jcp.2015.05.006 fatcat:3osr5b6wnrfybf3p3vfizzz5xm

A rapidly converging domain decomposition method for the Helmholtz equation

Christiaan C. Stolk
2013 Journal of Computational Physics  
We focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes, in combination with a certain ordering for solving the subdomain problems and a GMRES outer iteration  ...  Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces.  ...  Plots of the medium and a solution are given in Figure 2 . The size of the example varied between 100 3 and 400 3 (excluding PML layers on the sides).  ... 
doi:10.1016/j.jcp.2013.01.039 fatcat:q5gm7afnavginlrurajin4d57q

An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems

A. Bermúdez, L. Hervella-Nieto, A. Prieto, R. Rodríguez
2008 SIAM Journal on Scientific Computing  
The aim of this paper is to introduce an "exact" bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a non integrable absorbing function.  ...  In spite of the singularity of the absorbing function, the coupled fluid/PML problem is well posed when the solution is sought in an adequate weighted Sobolev space.  ...  Now, we prove the existence and uniqueness of smooth solutions of the following problem for the PML equation with Dirichlet data g: Findû ∈ W such that − div(A gradû) − γγk 2û = 0 in D, (7.1)û = g on S  ... 
doi:10.1137/060670912 fatcat:rsws45p6hnem5babqr66gasb5i

Perfectly matched layers for frequency-domain integral equation acoustic scattering problems

E J Alles, K W A van Dongen
2011 IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control  
In addition, it is shown that the integral equation method, both with and without PMLs, accurately reproduces pressure fields by comparing the obtained results with analytical solutions.  ...  The results demonstrate that strong attenuation (a factor of 200 in amplitude) of the scattered pressure field is achieved for thin layers with a thickness of less than a wavelength, and that the PMLs  ...  Validation of the Integral Equation Method The simplest scattering problem for which an analytical solution exists [21] is the one-dimensional case of a plane wave reflecting off a single interface between  ... 
doi:10.1109/tuffc.2011.1908 pmid:21622063 fatcat:wldmzxn5cvbfvmdeivg3wcsmli

Page 5185 of Mathematical Reviews Vol. , Issue 97H [page]

1997 Mathematical Reviews  
Plane wave solutions to the PML equations are developed and wave propagation and absorption properties are given.  ...  In- deed, if one sets a ~ |u| as usual, the artificial sound waves cease to exist when Ars is small and this adversely affects the iteration convergence rate.  ... 

Exponential convergence of Cartesian PML method for Maxwell's equations in a two-layer medium

Weiying Zheng, Xiaoqi Duan, Xue Jiang
2019 Mathematical Modelling and Numerical Analysis  
We prove the well-posedness of the PML truncated problem and that the PML solution converges exponentially to the exact solution as either the material parameter or the thickness of PML increases.  ...  To the best of the authors' knowledge, this is the first theoretical work on Cartesian PML method for Maxwell's equations in layered media.  ...  The second author is supported in part by China NSF grant 11771057 and 11671052.  ... 
doi:10.1051/m2an/2019082 fatcat:ywnnk4txfvb4rjmnhpjht3gjda

A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions

Lin Zschiedrich, Roland Klose, Achim Schädle, Frank Schmidt
2006 Journal of Computational and Applied Mathematics  
Numerical examples show exponential convergence behavior to the exact solution with the thickness of the PML sponge layer.  ...  In this paper we propose a new finite element realization of the Perfectly Matched Layer method (PML-method).  ...  We acknowledge support by the initiative DFG Research Center Matheon "Mathematics for key technologies" and by the German Federal Ministry of Education and Research, BMBF, under Contract No. 13N8252 ("  ... 
doi:10.1016/j.cam.2005.03.047 fatcat:qy3scrzz3jgepelugtbxyoj2ji

Analysis of a Cartesian PML approximation to acoustic scattering problems in R2

Seungil Kim, Joseph E. Pasciak
2010 Journal of Mathematical Analysis and Applications  
The exponential convergence of approximate solutions as a function of domain size is also shown.  ...  Our goal will be to extend the results of spherical/cylindrical PML to PML in Cartesian coordinates, that is, the well-posedness of Cartesian PML approximation on both the unbounded and truncated domains  ...  Acknowledgments This work was supported in part by the National Science Foundation through Grant DMS-0609544 and in part by award number KUS-C1-016-04 made by King Abdulla University of Science and Technology  ... 
doi:10.1016/j.jmaa.2010.05.006 fatcat:yh3ping4yzbfdkdgtxxhyp7psm

An Adaptive Finite Element PML Method for the Acoustic Scattering Problems in Layered Media

Xue Jiang, Yu Qi, Jianhua Yuan
2019 Communications in Computational Physics  
The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium.  ...  Numerical experiments are included to demonstrate the efficiency of the proposed method.  ...  The research of JY is supported by China NSF grants 11671052, 11471052 and 91630202.  ... 
doi:10.4208/cicp.oa-2018-0045 fatcat:4vjmpjxfkjgldgaj3uwxxp7qhm

Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems

Zhiming Chen, Xinming Wu
2012 SIAM Journal on Numerical Analysis  
It is proved in [21, 19, 10, 11] that the PML solution converges exponentially to the solution of the original scattering problem as the thickness of the PML layer tends to infinite.  ...  [29], and Gedney [16] for the Maxwell equations and a second order PML formulation in Grote and Sim [17] for wave equations.  ...  The authors would like to thank the referee for the constructive comments.  ... 
doi:10.1137/110835268 fatcat:4end3evi5ffjxn3dbzznv4bywq

On the Accuracy and Stability of the Perfectly Matched Layer in Transient Waveguides

Kenneth Duru, Gunilla Kreiss
2012 Journal of Scientific Computing  
In this paper we study the efficiency of a PML in a time-dependent waveguide governed by the scalar wave equation.  ...  Numerical stability and convergence of our numerical method follows.  ...  Convergence Let U,V,W,Ψ denote exact solutions of the continuous PML (14) and U,V,W,Ψ denote the restriction of the exact solutions of (14) on the grid.  ... 
doi:10.1007/s10915-012-9594-7 fatcat:3f6w3aufajd7nlnf35fivwohgm

A dimensionality reduction technique for 2D scattering problems in photonics

O V (Alyona) Ivanova, Remco Stoffer, Manfred Hammer
2010 Journal of Optics  
The technique reduces the spatial dimensionality of the problem by means of global, continuous mode expansion combined with a variational formalism; the resulting equations are solved using a finite element  ...  Transparent influx boundary conditions and perfectly matched layers are employed at the computational window boundaries. Numerical examples validate the method.  ...  Acknowledgements This work was supported by the Dutch Technology Foundation (BSIK / NanoNed project TOE.7143)  ... 
doi:10.1088/2040-8978/12/3/035502 fatcat:vc4cqaem3bdobj6ob5dbol6tbi
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