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Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation


Authors: Hélène Barucq, Théophile Chaumont-Frelet and Christian Gout
Journal: Math. Comp. 86 (2017), 2129-2157
MSC (2010): Primary 65N12, 65N15, 65N30; Secondary 35J05
DOI: https://doi.org/10.1090/mcom/3165
Published electronically: December 7, 2016
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Abstract: The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscillatory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.


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Additional Information

Hélène Barucq
Affiliation: INRIA Research Centre Bordeaux Sud-Ouest IPRA, University of Pau, IPRA, BP 1155, 64013 Pau, France
Email: helene.barucq@inria.fr

Théophile Chaumont-Frelet
Affiliation: INRIA Research Centre Bordeaux Sud-Ouest IPRA, University of Pau, IPRA, BP 1155, 64013 Pau, France – and – Normandie Université, INSA de Rouen, LMI, Av. de l’Université, 76801 St Etienne du Rouvray cedex, France
Email: theophile.chaumont_frelet@insa-rouen.fr

Christian Gout
Affiliation: Normandie Université, INSA de Rouen, LMI, Av. de l’Université, 76801 St Etienne du Rouvray cedex, France
Email: christian.gout@insa-rouen.fr

DOI: https://doi.org/10.1090/mcom/3165
Keywords: Helmholtz equation, highly heterogeneous media, high order method, medium approximation, stability estimates, depth imaging
Received by editor(s): July 8, 2014
Received by editor(s) in revised form: July 20, 2015, March 14, 2016, and March 16, 2016
Published electronically: December 7, 2016
Additional Notes: This work was partially supported by the project M2NUM (M2NUM is co-financed by the European Union with the European regional development fund (ERDF, HN0002137) and by the Normandie Regional Council) and the INRIA-TOTAL strategic action DIP. The authors also thank the Centre for Computer Resources of Normandy (CRIANN, \url{http://www.criann.fr}) where numerical simulations have been done (in parts)
Article copyright: © Copyright 2016 American Mathematical Society