SpringerBriefs in Mathematics

Symmetric Discontinuous Galerkin Methods for 1-D Waves

Fourier Analysis, Propagation, Observability and Applications

Authors: Marica, Aurora, Zuazua, Enrique

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About this book

This work describes the propagation properties of the so-called symmetric interior penalty discontinuous Galerkin (SIPG) approximations of the 1-d wave equation. This is done by means of linear approximations on uniform meshes. First, a careful Fourier analysis is constructed, highlighting the coexistence of two Fourier spectral branches or spectral diagrams (physical and spurious) related to the two components of the numerical solution (averages and jumps). Efficient filtering mechanisms are also developed by means of techniques previously proved to be appropriate for classical schemes like finite differences or P1-classical finite elements. In particular, the work presents a proof that the uniform observability property is recovered uniformly by considering initial data with null jumps and averages given by a bi-grid filtering algorithm. Finally, the book explains how these results can be extended to other more sophisticated conforming and non-conforming finite element methods, in particular to quadratic finite elements, local discontinuous Galerkin methods and a version of the SIPG method adding penalization on the normal derivatives of the numerical solution at the grid points.  This work is the first publication to contain a rigorous analysis of the discontinuous Galerkin methods for wave control problems. It will be of interest to a range of researchers specializing in wave approximations.

Table of contents (8 chapters)

  • Preliminaries

    Marica, Aurora (et al.)

    Pages 1-13

  • Discontinuous Galerkin Approximations and Main Results

    Marica, Aurora (et al.)

    Pages 15-25

  • Bibliographical Notes

    Marica, Aurora (et al.)

    Pages 27-29

  • Fourier Analysis of the Discontinuous Galerkin Methods

    Marica, Aurora (et al.)

    Pages 31-39

  • On the Lack of Uniform Observability for Discontinuous Galerkin Approximations of Waves

    Marica, Aurora (et al.)

    Pages 41-50

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-1-4614-5811-1
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA
  • ISBN 978-1-4614-5810-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Symmetric Discontinuous Galerkin Methods for 1-D Waves
Book Subtitle
Fourier Analysis, Propagation, Observability and Applications
Authors
Series Title
SpringerBriefs in Mathematics
Copyright
2014
Publisher
Springer-Verlag New York
Copyright Holder
Aurora Marica, Enrique Zuazua
eBook ISBN
978-1-4614-5811-1
DOI
10.1007/978-1-4614-5811-1
Softcover ISBN
978-1-4614-5810-4
Series ISSN
2191-8198
Edition Number
1
Number of Pages
XVI, 104
Number of Illustrations and Tables
18 illustrations in colour
Topics