Mathématiques et Applications

Mathematical Aspects of Discontinuous Galerkin Methods

Authors: Di Pietro, Daniele Antonio, Ern, Alexandre

  • Understanding the mathematical foundations helps the reader design methods for new applications 
  • Bridging the gap between finite volumes, finite elements, and discontinuous Galerkin methods provides new insight on numerical methods
  • The mathematical setting for the continuous model is a key to successful approximation methods
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About this book

This book introduces the basic ideas to build discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. The presentation is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite element and finite volume viewpoints are exploited to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed.

Reviews

From the reviews:

“The goal of this book is to provide graduate students and researchers in numerical methods with the basic mathematical concepts to design and analyze discontinuous Galerkin (DG) methods for various model problems, starting at an introductory level and further elaborating on more advanced topics, considering that DG methods have tremendously developed in the last decade.” (Rémi Vaillancourt, Mathematical Reviews, January, 2013)

“The book is structured in three parts: scalar first order PDEs, scalar second order PDEs, and systems. … For researchers in numerical analysis it is nice to see that for all problem classes the authors start with a full analysis of existence, uniqueness, and properties of the solution of the continuous problem. … this new monograph is an extremely valuable source concerning the theoretical function of dG methods for the advanced reader.” (H.-G. Roos, SIAM Review, Vol. 55 (2), 2013)

“This new monograph is an extremely valuable collection of the mathematical treatment of discontinuous Galerkin methods with 300 references and providing profound insight into the required techniques. It collects and presents also several recent results for elliptic and non-elliptic, stationary and non-stationary partial differential equations in a unified framework. Thus it is strongly recommendable for researchers in the field.” (Christian Wieners, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 92 (7), 2012)

“The aim of the book is ‘to provide the reader with the basic mathematical concepts to design and analyze discontinuous Galerkin methods for various model problems, starting at an introductory level and further elaborating on more advanced topics’. … Some useful practical implementation aspects are considered in an Appendix. The bibliography contains more than 300 entries.” (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1231, 2012)


Table of contents (7 chapters)

  • Basic Concepts

    Pietro, Daniele Antonio (et al.)

    Pages 1-34

  • Steady Advection-Reaction

    Pietro, Daniele Antonio (et al.)

    Pages 37-65

  • Unsteady First-Order PDEs

    Pietro, Daniele Antonio (et al.)

    Pages 67-115

  • PDEs with Diffusion

    Pietro, Daniele Antonio (et al.)

    Pages 119-186

  • Additional Topics on Pure Diffusion

    Pietro, Daniele Antonio (et al.)

    Pages 187-237

Buy this book

eBook $89.00
price for USA (gross)
  • ISBN 978-3-642-22980-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $124.00
price for USA
  • ISBN 978-3-642-22979-4
  • 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
Mathematical Aspects of Discontinuous Galerkin Methods
Authors
Series Title
Mathématiques et Applications
Series Volume
69
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-22980-0
DOI
10.1007/978-3-642-22980-0
Softcover ISBN
978-3-642-22979-4
Series ISSN
1154-483X
Edition Number
1
Number of Pages
XVII, 384
Number of Illustrations and Tables
34 b/w illustrations
Topics