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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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    Authors
    Zitelli, J.
    Muga, I.
    Demkowicz, L.
    Gopalakrishnan, J.
    Pardo, D.
    Calo, Victor
    Date
    2011
    Type
    Journal Article
    
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    Citation
    Zitelli, J. and Muga, I. and Demkowicz, L. and Gopalakrishnan, J. and Pardo, D. and Calo, V. 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): pp. 2406-2432.
    Source Title
    Journal of Computational Physics
    DOI
    10.1016/j.jcp.2010.12.001
    ISSN
    0021-9991
    School
    Department of Applied Geology
    URI
    http://hdl.handle.net/20.500.11937/51523
    Collection
    • Curtin Research Publications
    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.

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