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Article
A Class of Discontinuous Petrov–Galerkin Methods. Part IV: The Optimal Test Norm and Time-Harmonic Wave Propagation in 1D.
Journal of Computational Physics
  • Jeffrey Zitelli, University of Texas at Austin
  • Leszek Demkowicz, University of Texas at Austin
  • Jay Gopalakrishnan, Portland State University
  • D. Pardo, University of the Basque Country
  • V. M. Calo, King Abdullah University of Science and Technology
Document Type
Post-Print
Publication Date
10-1-2010
Subjects
  • Galerkin methods,
  • Harmonics (Electric waves),
  • Helmholtz equation,
  • Finite element method
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 L 2-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.
Description

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics, Volume 230, Issue 7 (2011).

DOI
10.1016/j.jcp.2010.12.001
Persistent Identifier
http://archives.pdx.edu/ds/psu/10629
Citation Information
Jeffrey Zitelli, Leszek Demkowicz, Jay Gopalakrishnan, D. Pardo, et al.. "A Class of Discontinuous Petrov–Galerkin Methods. Part IV: The Optimal Test Norm and Time-Harmonic Wave Propagation in 1D." Journal of Computational Physics (2010)
Available at: http://works.bepress.com/jay-gopalakrishnan/56/