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© 2012

Mathematical Aspects of Discontinuous Galerkin Methods

Benefits

  • 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

Book

Part of the Mathématiques et Applications book series (MATHAPPLIC, volume 69)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Daniele Antonio Di Pietro, Alexandre Ern
    Pages 1-34
  3. Scalar First-Order PDEs

    1. Front Matter
      Pages 35-35
    2. Daniele Antonio Di Pietro, Alexandre Ern
      Pages 37-65
    3. Daniele Antonio Di Pietro, Alexandre Ern
      Pages 67-115
  4. Scalar Second-Order PDEs

    1. Front Matter
      Pages 117-117
    2. Daniele Antonio Di Pietro, Alexandre Ern
      Pages 119-186
    3. Daniele Antonio Di Pietro, Alexandre Ern
      Pages 187-237
  5. Systems

    1. Front Matter
      Pages 239-239
    2. Daniele Antonio Di Pietro, Alexandre Ern
      Pages 241-291
    3. Daniele Antonio Di Pietro, Alexandre Ern
      Pages 293-341
  6. Back Matter
    Pages 343-384

About this book

Introduction

This book introduces the basic ideas for building discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. It 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 utilized 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.

Keywords

Discontinuous Galerkin methods First-order PDEs Friedrichs' systems Incompressible Navier-Stokes equations Second-order PDEs

Authors and affiliations

  1. 1., Department of Applied MathematicsIFP Energies nouvellesRueil-MalmaisonFrance
  2. 2., CERMICS, Ecole des Ponts ParisTechUniversité Paris EstMarne la Vallée cedex 2France

Bibliographic information

Industry Sectors
Energy, Utilities & Environment
Engineering

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)