Symmetric Discontinuous Galerkin Methods for 1-D Waves

Fourier Analysis, Propagation, Observability and Applications

  • Aurora Marica
  • Enrique Zuazua

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Aurora Marica, Enrique Zuazua
    Pages 1-13
  3. Aurora Marica, Enrique Zuazua
    Pages 15-25
  4. Aurora Marica, Enrique Zuazua
    Pages 27-29
  5. Aurora Marica, Enrique Zuazua
    Pages 31-39
  6. Aurora Marica, Enrique Zuazua
    Pages 51-81
  7. Aurora Marica, Enrique Zuazua
    Pages 83-91
  8. Aurora Marica, Enrique Zuazua
    Pages 93-95
  9. Back Matter
    Pages 97-104

About this book

Introduction

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.

Keywords

discontinuous Galerkin approximation linear approximation non-uniform propagation properties physical/spurious dispersion relation uniform grid wave equation

Authors and affiliations

  • Aurora Marica
    • 1
  • Enrique Zuazua
    • 2
  1. 1.Institute of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  2. 2.BCAM-Basque Center for Applied MathematiBilbaoSpain

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-5811-1
  • Copyright Information Aurora Marica, Enrique Zuazua 2014
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-5810-4
  • Online ISBN 978-1-4614-5811-1
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • About this book
Industry Sectors
Aerospace
Oil, Gas & Geosciences