Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering
Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering
View/Date of Publication
03.2015Publisher
Institut für Numerische SimulationPublisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11868
Abstract
We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ as a variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous Q1 finite elements at a coarse discretization scale H as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale h. The diameter of the support of the test functions behaves like mH for some oversampling parameter m. Provided m is of the order of log(κ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ−1. In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations. Therefore, the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes. We present numerical experiments in two and three space dimensions.
Subjects
multiscale method, pollution effect, wave propagation, Helmholtz problem, finite element method
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1016/j.cma.2015.06.017Part of Series
INS Preprints ; 1504Gallistl, Dietmar; Peterseim, Daniel: Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering. In: INS Preprints, 1504.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11868
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11868
@unpublished{handle:20.500.11811/11868,
author = {{Dietmar Gallistl} and {Daniel Peterseim}},
title = {Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering},
publisher = {Institut für Numerische Simulation},
year = 2015,
month = mar,
INS Preprints},
volume = 1504,
note = {We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ as a variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous Q1 finite elements at a coarse discretization scale H as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale h. The diameter of the support of the test functions behaves like mH for some oversampling parameter m. Provided m is of the order of log(κ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ−1. In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations. Therefore, the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes. We present numerical experiments in two and three space dimensions.},
url = {https://hdl.handle.net/20.500.11811/11868}
}
author = {{Dietmar Gallistl} and {Daniel Peterseim}},
title = {Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering},
publisher = {Institut für Numerische Simulation},
year = 2015,
month = mar,
INS Preprints},
volume = 1504,
note = {We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ as a variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous Q1 finite elements at a coarse discretization scale H as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale h. The diameter of the support of the test functions behaves like mH for some oversampling parameter m. Provided m is of the order of log(κ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ−1. In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations. Therefore, the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes. We present numerical experiments in two and three space dimensions.},
url = {https://hdl.handle.net/20.500.11811/11868}
}
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