TY - JOUR

T1 - Analysis of a Cartesian PML approximation to acoustic scattering problems in and

AU - Bramble, James H.

AU - Pasciak, Joseph E.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: This work was supported in part by award number KUS-C1-016-04 made by King Abdulla University of Science and Technology (KAUST). It was also partially supported by National Science Foundation grant number DMS-1216551.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2013/8

Y1 - 2013/8

N2 - We consider the application of a perfectly matched layer (PML) technique applied in Cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift ("stretching") and leads to a variable complex coefficient equation for the acoustic wave posed on an infinite domain, the complement of the bounded scatterer. The use of Cartesian geometry leads to a PML operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML reformulation results in a problem whose solution coincides with the original solution inside the PML layer while decaying exponentially outside. The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). This paper provides new stability estimates for the Cartesian PML approximations both on the infinite and the truncated domain. We first investigate the stability of the infinite PML approximation as a function of the PML strength σ0. This is done for PML methods which involve continuous piecewise smooth stretching as well as piecewise constant stretching functions. We next introduce a truncation parameter M which determines the size of the PML layer. Our analysis shows that the truncated PML problem is stable provided that the product of Mσ0 is sufficiently large, in which case the solution of the problem on the truncated domain converges exponentially to that of the original problem in the domain of interest near the scatterer. This justifies the simple computational strategy of selecting a fixed PML layer and increasing σ0 to obtain the desired accuracy. The results of numerical experiments varying M and σ0 are given which illustrate the theoretically predicted behavior. © 2013 Elsevier B.V. All rights reserved.

AB - We consider the application of a perfectly matched layer (PML) technique applied in Cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift ("stretching") and leads to a variable complex coefficient equation for the acoustic wave posed on an infinite domain, the complement of the bounded scatterer. The use of Cartesian geometry leads to a PML operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML reformulation results in a problem whose solution coincides with the original solution inside the PML layer while decaying exponentially outside. The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). This paper provides new stability estimates for the Cartesian PML approximations both on the infinite and the truncated domain. We first investigate the stability of the infinite PML approximation as a function of the PML strength σ0. This is done for PML methods which involve continuous piecewise smooth stretching as well as piecewise constant stretching functions. We next introduce a truncation parameter M which determines the size of the PML layer. Our analysis shows that the truncated PML problem is stable provided that the product of Mσ0 is sufficiently large, in which case the solution of the problem on the truncated domain converges exponentially to that of the original problem in the domain of interest near the scatterer. This justifies the simple computational strategy of selecting a fixed PML layer and increasing σ0 to obtain the desired accuracy. The results of numerical experiments varying M and σ0 are given which illustrate the theoretically predicted behavior. © 2013 Elsevier B.V. All rights reserved.

UR - http://hdl.handle.net/10754/597548

UR - https://linkinghub.elsevier.com/retrieve/pii/S0377042713000149

UR - http://www.scopus.com/inward/record.url?scp=84873944446&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2012.12.022

DO - 10.1016/j.cam.2012.12.022

M3 - Article

VL - 247

SP - 209

EP - 230

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -