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

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    Type
    Article
    Authors
    Zitelli, J.
    Muga, Ignacio
    Demkowicz, Leszek F.
    Gopalakrishnan, Jayadeep
    Pardo, David
    Calo, Victor M. cc
    KAUST Department
    Applied Mathematics and Computational Science Program
    Earth Science and Engineering Program
    Environmental Science and Engineering Program
    Numerical Porous Media SRI Center (NumPor)
    Physical Science and Engineering (PSE) Division
    Date
    2011-04
    Permanent link to this record
    http://hdl.handle.net/10754/561739
    
    Metadata
    Show full item record
    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.
    Citation
    Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D., & Calo, V. M. (2011). A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D. Journal of Computational Physics, 230(7), 2406–2432. doi:10.1016/j.jcp.2010.12.001
    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.
    Publisher
    Elsevier BV
    Journal
    Journal of Computational Physics
    DOI
    10.1016/j.jcp.2010.12.001
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.jcp.2010.12.001
    Scopus Count
    Collections
    Articles; Environmental Science and Engineering Program; Applied Mathematics and Computational Science Program; Physical Science and Engineering (PSE) Division; Earth Science and Engineering Program

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