A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

Handle URI:
http://hdl.handle.net/10754/561739
Title:
A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D
Authors:
Zitelli, J.; Muga, Ignacio; Demkowicz, Leszek F.; Gopalakrishnan, Jayadeep; Pardo, David; Calo, Victor M. ( 0000-0002-1805-4045 )
Abstract:
The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed. © 2010 Elsevier Inc.
KAUST Department:
Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division; Environmental Science and Engineering Program; Numerical Porous Media SRI Center (NumPor)
Publisher:
Elsevier
Journal:
Journal of Computational Physics
Issue Date:
Apr-2011
DOI:
10.1016/j.jcp.2010.12.001
Type:
Article
ISSN:
00219991
Sponsors:
J. Zitelli was supported by an ONR Graduate Traineeship and CAM Fellowhip. I. Muga was supported by Sistema Bicentenario BECAS CHILE (Chilean Government). L. Demkowicz was supported by a Collaborative Research Grant from King Abdullah University of Science and Technology (KAUST). J. Gopalakrishnan was supported by the National Science Foundation under Grant No. DMS-1014817.
Appears in Collections:
Articles; Environmental Science and Engineering Program; Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorZitelli, J.en
dc.contributor.authorMuga, Ignacioen
dc.contributor.authorDemkowicz, Leszek F.en
dc.contributor.authorGopalakrishnan, Jayadeepen
dc.contributor.authorPardo, Daviden
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2015-08-03T09:03:31Zen
dc.date.available2015-08-03T09:03:31Zen
dc.date.issued2011-04en
dc.identifier.issn00219991en
dc.identifier.doi10.1016/j.jcp.2010.12.001en
dc.identifier.urihttp://hdl.handle.net/10754/561739en
dc.description.abstractThe phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed. © 2010 Elsevier Inc.en
dc.description.sponsorshipJ. Zitelli was supported by an ONR Graduate Traineeship and CAM Fellowhip. I. Muga was supported by Sistema Bicentenario BECAS CHILE (Chilean Government). L. Demkowicz was supported by a Collaborative Research Grant from King Abdullah University of Science and Technology (KAUST). J. Gopalakrishnan was supported by the National Science Foundation under Grant No. DMS-1014817.en
dc.publisherElsevieren
dc.subjectDiscontinuous Petrov Galerkinen
dc.subjectDispersionen
dc.subjectDPGen
dc.subjectHelmholtzen
dc.subjectHigh frequencyen
dc.subjectPhase erroren
dc.subjectRobustnessen
dc.subjectTime harmonicen
dc.subjectWave propagationen
dc.titleA class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1Den
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentEnvironmental Science and Engineering Programen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalJournal of Computational Physicsen
dc.contributor.institutionInstitute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712, United Statesen
dc.contributor.institutionInstituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chileen
dc.contributor.institutionDepartment of Mathematics, University of Florida, Gainesville, FL 32611-8105, United Statesen
dc.contributor.institutionDepartment of Applied Mathematics, Statistics and Operational Research, Univ. of the Basque Country (UPV/EHU), Ikerbasque (Basque Foundation for Sci.), Bilbao, Spainen
kaust.authorCalo, Victor M.en
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